JEE Main & Advanced Mathematics Conic Sections Question Bank

Ellipse Total Questions - 108

1)

If the latus rectum of an ellipse be equal to half of its minor axis, then its eccentricity is [MP PET 1991, 97; Karnataka CET 2000]

A)
           3/2

B)
           \[\sqrt{3}/2\]

C)
           2/3

D)
           \[\sqrt{2}/3\]
View Solution
2)

If distance between the directrices be thrice the distance between the foci, then eccentricity of ellipse is

A)
           1/2

B)
           2/3

C)
           \[1/\sqrt{3}\]

D)
           4/5
View Solution
3)

The equation of the ellipse whose centre is at origin and which passes through the points (?3, 1) and (2, ?2) is

A)
           \[5{{x}^{2}}+3{{y}^{2}}=32\]

B)
           \[3{{x}^{2}}+5{{y}^{2}}=32\]

C)
           \[5{{x}^{2}}-3{{y}^{2}}=32\]

D)
           \[3{{x}^{2}}+5{{y}^{2}}+32=0\]
View Solution
4)

If the eccentricity of an ellipse be 5/8 and the distance between its foci be 10, then its latus rectum is

A)
           39/4

B)
           12

C)
           15

D)
           37/2
View Solution
5)

If the foci and vertices of an ellipse be \[(\pm 1,\ 0)\] and \[(\pm 2,\ 0)\], then the minor axis of the ellipse is

A)
           \[2\sqrt{5}\]

B)
           2

C)
           4

D)
           \[2\sqrt{3}\]
View Solution
6)

The equations of the directrices of the ellipse \[16{{x}^{2}}+25{{y}^{2}}=400\] are

A)
           \[2x=\pm 25\]

B)
           \[5x=\pm 9\]

C)
           \[3x=\pm 10\]

D)
           None of these
View Solution
7)

The eccentricity of an ellipse is 2/3, latus rectum is 5 and centre is (0, 0). The equation of the ellipse is

A)
           \[\frac{{{x}^{2}}}{81}+\frac{{{y}^{2}}}{45}=1\]

B)
           \[\frac{4{{x}^{2}}}{81}+\frac{4{{y}^{2}}}{45}=1\]

C)
           \[\frac{{{x}^{2}}}{9}+\frac{{{y}^{2}}}{5}=1\]

D)
           \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\]
View Solution
8)

The latus rectum of an ellipse is 10 and the minor axis is equal to the distance between the foci. The equation of the ellipse is

A)
           \[{{x}^{2}}+2{{y}^{2}}=100\]

B)
           \[{{x}^{2}}+\sqrt{2}{{y}^{2}}=10\]

C)
           \[{{x}^{2}}-2{{y}^{2}}=100\]

D)
           None of these
View Solution
9)

The distance between the directrices of the ellipse \[\frac{{{x}^{2}}}{36}+\frac{{{y}^{2}}}{20}=1\] is

A)
           8

B)
           12

C)
           18

D)
           24
View Solution
10)

The distance between the foci of the ellipse \[3{{x}^{2}}+4{{y}^{2}}=48\] is

A)
           2

B)
           4

C)
           6

D)
           8
View Solution
11)

The equation of the ellipse whose vertices are \[(\pm 5,\ 0)\] and foci are \[(\pm 4,\ 0)\] is

A)
           \[9{{x}^{2}}+25{{y}^{2}}=225\]

B)
           \[25{{x}^{2}}+9{{y}^{2}}=225\]

C)
           \[3{{x}^{2}}+4{{y}^{2}}=192\]

D)
           None of these
View Solution
12)

The equation of the ellipse whose foci are \[(\pm 5,\ 0)\] and one of its directrix is \[5x=36\], is

A)
           \[\frac{{{x}^{2}}}{36}+\frac{{{y}^{2}}}{11}=1\]

B)
           \[\frac{{{x}^{2}}}{6}+\frac{{{y}^{2}}}{\sqrt{11}}=1\]

C)
           \[\frac{{{x}^{2}}}{6}+\frac{{{y}^{2}}}{11}=1\]

D)
           None of these
View Solution
13)

If the eccentricity of an ellipse be \[1/\sqrt{2}\], then its latus rectum is equal to its

A)
           Minor axis

B)
           Semi-minor axis

C)
           Major axis

D)
           Semi-major axis
View Solution
14)

The length of the latus rectum of the ellipse \[5{{x}^{2}}+9{{y}^{2}}=45\] is      [MNR 1978, 80, 81]

A)
           \[\sqrt{5}/4\]

B)
           \[\sqrt{5}/2\]

C)
           5/3

D)
           10/3
View Solution
15)

If the distance between a focus and corresponding directrix of an ellipse be 8 and the eccentricity be 1/2, then length of the minor axis is

A)
           3

B)
           \[4\sqrt{2}\]

C)
           6

D)
           None of these
View Solution
16)

Eccentricity of the conic \[16{{x}^{2}}+7{{y}^{2}}=112\]is [MNR 1981]

A)
           \[3/\sqrt{7}\]

B)
           7/16

C)
           3/4

D)
           4/3
View Solution
17)

If the distance between the foci of an ellipse be equal to its minor axis, then its eccentricity is

A)
           1/2

B)
           \[1/\sqrt{2}\]

C)
           1/3

D)
           \[1/\sqrt{3}\]
View Solution
18)

An ellipse passes through the point (?3, 1) and its eccentricity is \[\sqrt{\frac{2}{5}}\]. The equation of the ellipse is

A)
           \[3{{x}^{2}}+5{{y}^{2}}=32\]

B)
           \[3{{x}^{2}}+5{{y}^{2}}=25\]

C)
           \[3{{x}^{2}}+{{y}^{2}}=4\]

D)
           \[3{{x}^{2}}+{{y}^{2}}=9\]
View Solution
19)

The lengths of major and minor axis of an ellipse are 10 and 8 respectively and its major axis along the y-axis. The equation of the ellipse referred to its centre as origin is [Pb. CET 2003]

A)
           \[\frac{{{x}^{2}}}{25}+\frac{{{y}^{2}}}{16}=1\]

B)
           \[\frac{{{x}^{2}}}{16}+\frac{{{y}^{2}}}{25}=1\]

C)
           \[\frac{{{x}^{2}}}{100}+\frac{{{y}^{2}}}{64}=1\]

D)
           \[\frac{{{x}^{2}}}{64}+\frac{{{y}^{2}}}{100}=1\]
View Solution
20)

If the centre, one of the foci and semi-major axis of an ellipse be (0, 0), (0, 3) and 5 then its equation is [AMU 1981]

A)
           \[\frac{{{x}^{2}}}{16}+\frac{{{y}^{2}}}{25}=1\]

B)
           \[\frac{{{x}^{2}}}{25}+\frac{{{y}^{2}}}{16}=1\]

C)
           \[\frac{{{x}^{2}}}{9}+\frac{{{y}^{2}}}{25}=1\]

D)
           None of these
View Solution
21)

The equation of the ellipse whose one of the vertices is (0,7) and the corresponding directrix is \[y=12\], is

A)
           \[95{{x}^{2}}+144{{y}^{2}}=4655\]

B)
           \[144{{x}^{2}}+95{{y}^{2}}=4655\]

C)
           \[95{{x}^{2}}+144{{y}^{2}}=13680\]

D)
           None of these
View Solution
22)

The equation \[2{{x}^{2}}+3{{y}^{2}}=30\] represents     [MP PET 1988]

A)
           A circle

B)
           An ellipse

C)
           A hyperbola

D)
           A parabola
View Solution
23)

hThe equation of the ellipse whose latus rectum is 8 and whose eccentricity is \[\frac{1}{\sqrt{2}}\], referred to the principal axes of coordinates, is [MP PET 1993]

A)
           \[\frac{{{x}^{2}}}{18}+\frac{{{y}^{2}}}{32}=1\]

B)
           \[\frac{{{x}^{2}}}{8}+\frac{{{y}^{2}}}{9}=1\]

C)
           \[\frac{{{x}^{2}}}{64}+\frac{{{y}^{2}}}{32}=1\]

D)
           \[\frac{{{x}^{2}}}{16}+\frac{{{y}^{2}}}{24}=1\]
View Solution
24)

Eccentricity of the ellipse whose latus rectum is equal to the distance between two focus points, is

A)
           \[\frac{\sqrt{5}+1}{2}\]

B)
           \[9{{x}^{2}}+5{{y}^{2}}-30y=0\]

C)
           \[\frac{\sqrt{5}}{2}\]

D)
           \[\frac{\sqrt{3}}{2}\]
View Solution
25)

For the ellipse \[3{{x}^{2}}+4{{y}^{2}}=12\], the length of latus rectum is          [MNR 1973]

A)
           \[\frac{3}{2}\]

B)
           3

C)
           \[\frac{8}{3}\]

D)
           \[\sqrt{\frac{3}{2}}\]
View Solution
26)

For the ellipse \[\frac{{{x}^{2}}}{64}+\frac{{{y}^{2}}}{28}=1\], the eccentricity is               [MNR 1974]

A)
           \[\frac{3}{4}\]

B)
           \[\frac{4}{3}\]

C)
           \[\frac{1}{\sqrt{7}}\]

D)
           \[\frac{{{x}^{2}}}{4}+{{y}^{2}}=1\]
View Solution
27)

If the length of the major axis of an ellipse is three times the length of its minor axis, then its eccentricity is [EAMCET 1990]

A)
           \[\frac{1}{3}\]

B)
           \[\frac{1}{\sqrt{3}}\]

C)
           \[\frac{1}{\sqrt{2}}\]

D)
           \[\frac{2\sqrt{2}}{3}\]
View Solution
28)

The length of the latus rectum of an ellipse is \[\frac{1}{3}\] of the major axis. Its eccentricity is [EAMCET 1991]

A)
           \[\frac{2}{3}\]

B)
           \[\sqrt{\frac{2}{3}}\]

C)
           \[\frac{5\times 4\times 3}{{{7}^{3}}}\]

D)
           \[{{\left( \frac{3}{4} \right)}^{4}}\]
View Solution
29)

An ellipse is described by using an endless string which is passed over two pins. If the axes are 6 cm and 4 cm, the necessary length of the string and the distance between the pins respectively in cm, are             [MNR 1989]

A)
           \[6,\ 2\sqrt{5}\]

B)
           \[6,\ \sqrt{5}\]

C)
           \[4,\ 2\sqrt{5}\]

D)
           None of these
View Solution
30)

The equation \[\frac{{{x}^{2}}}{2-r}+\frac{{{y}^{2}}}{r-5}+1=0\] represents an ellipse, if [MP PET 1995]

A)
           \[r>2\]

B)
           \[2<r<5\]

C)
           \[r>5\]

D)
           None of these
View Solution
31)

The locus of the point of intersection of perpendicular tangents to the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\], is               [MP PET 1995]

A)
           \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}-{{b}^{2}}\]

B)
           \[{{x}^{2}}-{{y}^{2}}={{a}^{2}}-{{b}^{2}}\]

C)
           \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}+{{b}^{2}}\]

D)
           \[{{x}^{2}}-{{y}^{2}}={{a}^{2}}+{{b}^{2}}\]
View Solution
32)

The length of the latus rectum of the ellipse \[\frac{{{x}^{2}}}{36}+\frac{{{y}^{2}}}{49}=1\] [Karnataka CET 1993]

A)
           98/6

B)
           72/7

C)
           72/14

D)
           98/12
View Solution
33)

The distance of the point \['\theta '\]on the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] from a focus is

A)
           \[a(e+\cos \theta )\]

B)
           \[a(e-\cos \theta )\]

C)
           \[a(1+e\cos \theta )\]

D)
           \[a(1+2e\cos \theta )\]
View Solution
34)

The equation of the ellipse whose one focus is at (4, 0) and whose eccentricity is 4/5, is                 [Karnataka CET 1993]

A)
           \[\frac{{{x}^{2}}}{{{3}^{2}}}+\frac{{{y}^{2}}}{{{5}^{2}}}=1\]

B)
           \[\frac{{{x}^{2}}}{{{5}^{2}}}+\frac{{{y}^{2}}}{{{3}^{2}}}=1\]

C)
           \[\frac{{{x}^{2}}}{{{5}^{2}}}+\frac{{{y}^{2}}}{{{4}^{2}}}=1\]

D)
           \[\frac{{{x}^{2}}}{{{4}^{2}}}+\frac{{{y}^{2}}}{{{5}^{2}}}=1\]
View Solution
35)

The foci of \[16{{x}^{2}}+25{{y}^{2}}=400\] are          [BIT Ranchi 1996]

A)
           \[(\pm 3,\ 0)\]

B)
           \[(0,\ \pm 3)\]

C)
           \[(3,\ -3)\]

D)
           \[(-3,\ 3)\]
View Solution
36)

Eccentricity of the ellipse \[9{{x}^{2}}+25{{y}^{2}}=225\] is [Kerala (Engg.) 2002]

A)
           \[\frac{3}{5}\]

B)
           \[\frac{4}{5}\]

C)
           \[\frac{9}{25}\]

D)
           \[\frac{\sqrt{34}}{5}\]
View Solution
37)

The eccentricity of the ellipse \[25{{x}^{2}}+16{{y}^{2}}=100\], is

A)
           \[\frac{5}{14}\]

B)
           \[\frac{4}{5}\]

C)
           \[\frac{3}{5}\]

D)
           \[\frac{2}{5}\]
View Solution
38)

The length of the latus rectum of the ellipse \[9{{x}^{2}}+4{{y}^{2}}=1\], is       [MP PET 1999]

A)
           \[\frac{3}{2}\]

B)
           \[\frac{8}{3}\]

C)
           \[\frac{4}{9}\]

D)
           \[\frac{8}{9}\]
View Solution
39)

The locus of a variable point whose distance from (?2, 0) is \[\frac{2}{3}\] times its distance from the line \[x=-\frac{9}{2}\], is                [IIT 1994]

A)
           Ellipse

B)
           Parabola

C)
           Hyperbola

D)
           None of these
View Solution
40)

If \[P\equiv (x,\ y)\], \[{{F}_{1}}\equiv (3,\ 0)\], \[{{F}_{2}}\equiv (-3,\ 0)\] and \[16{{x}^{2}}+25{{y}^{2}}=400\], then \[P{{F}_{1}}+P{{F}_{2}}\] equals                                                                                                    [IIT 1998]

A)
           8

B)
           6

C)
           10

D)
           12
View Solution
41)

P is any point on the ellipse\[9{{x}^{2}}+36{{y}^{2}}=324\]., whose foci are S and S?. Then \[SP+S'P\] equals [DCE 1999]

A)
           3

B)
           12

C)
           36

D)
           324
View Solution
42)

What is the equation of the ellipse with foci \[(\pm 2,\ 0)\] and eccentricity \[=\frac{1}{2}\]          [DCE 1999]

A)
           \[3{{x}^{2}}+4{{y}^{2}}=48\]

B)
           \[4{{x}^{2}}+3{{y}^{2}}=48\]

C)
           \[3{{x}^{2}}+4{{y}^{2}}=0\]

D)
           \[4{{x}^{2}}+3{{y}^{2}}=0\]
View Solution
43)

The eccentricity of the ellipse \[4{{x}^{2}}+9{{y}^{2}}=36\], is [MP PET 2000]

A)
           \[\frac{1}{2\sqrt{3}}\]

B)
           \[\frac{1}{\sqrt{3}}\]

C)
           \[\frac{\sqrt{5}}{3}\]

D)
           \[\frac{\sqrt{5}}{6}\]
View Solution
44)

The eccentricity of the ellipse \[25{{x}^{2}}+16{{y}^{2}}=400\] is [MP PET 2001]

A)
           3/5

B)
           1/3

C)
           2/5

D)
           1/5
View Solution
45)

The distance between the foci of an ellipse is 16 and eccentricity is \[\frac{1}{2}\]. Length of the major axis of the ellipse is [Karnataka CET 2001]

A)
           8

B)
           64

C)
           16

D)
           32
View Solution
46)

If the eccentricity of the two ellipse \[\frac{{{x}^{2}}}{169}+\frac{{{y}^{2}}}{25}=1\]and \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] are equal, then the value of \[a/b\] is [UPSEAT 2001]

A)
           5/13

B)
           6/13

C)
           13/5

D)
           13/6
View Solution
47)

In the ellipse, minor axis is 8 and eccentricity is \[\frac{\sqrt{5}}{3}\]. Then major axis is                [Karnataka CET 2002]

A)
           6

B)
           12

C)
           10

D)
           16
View Solution
48)

In an ellipse \[9{{x}^{2}}+5{{y}^{2}}=45\], the distance between the foci is        [Karnataka CET 2002]

A)
           \[4\sqrt{5}\]

B)
           \[\frac{49}{4}{{x}^{2}}-\frac{51}{196}{{y}^{2}}=1\]

C)
           3

D)
           4
View Solution
49)

Equation of the ellipse with eccentricity \[\frac{1}{2}\] and foci at \[(\pm 1,\ 0)\] is        [MP PET 2002]

A)
           \[\frac{{{x}^{2}}}{3}+\frac{{{y}^{2}}}{4}=1\]

B)
           \[\frac{{{x}^{2}}}{4}+\frac{{{y}^{2}}}{3}=1\]

C)
           \[\frac{{{x}^{2}}}{3}+\frac{{{y}^{2}}}{4}=\frac{4}{3}\]

D)
           None of these
View Solution
50)

The sum of focal distances of any point on the ellipse with major and minor axes as 2a and 2b respectively, is equal to               [MP PET 2003]

A)
           2a

B)
           \[\frac{2a}{b}\]

C)
           \[\frac{2b}{a}\]

D)
           \[\frac{{{b}^{2}}}{a}\]
View Solution
51)

The equation of ellipse whose distance between the foci is equal to 8 and distance between the directrix is 18, is

A)
           \[5{{x}^{2}}-9{{y}^{2}}=180\]

B)
           \[9{{x}^{2}}+5{{y}^{2}}=180\]

C)
           \[{{x}^{2}}+9{{y}^{2}}=180\]

D)
           \[5{{x}^{2}}+9{{y}^{2}}=180\]
View Solution
52)

In an ellipse the distance between its foci is 6 and its minor axis is 8. Then its eccentricity is          [EAMCET 1994]

A)
           \[\frac{4}{5}\]

B)
           \[\frac{1}{\sqrt{52}}\]

C)
           \[\frac{3}{5}\]

D)
           \[25{{x}^{2}}+144{{y}^{2}}=900\]
View Solution
53)

If a bar of given length moves with its extremities on two fixed straight lines at right angles, then the locus of any point on bar marked on the bar describes a/an                                                                [Orissa JEE 2003]

A)
           Circle

B)
           Parabola

C)
           Ellipse

D)
           Hyperbola
View Solution
54)

The centre of the ellipse \[4{{x}^{2}}+9{{y}^{2}}-16x-54y+61=0\] is [MP PET 1992]

A)
           (1, 3)

B)
           (2, 3)

C)
           (3, 2)

D)
           (3, 1)
View Solution
55)

Latus rectum of ellipse \[4{{x}^{2}}+9{{y}^{2}}-8x-36y+4=0\] is [MP PET 1989]

A)
           8/3

B)
           4/3

C)
           \[\frac{\sqrt{5}}{3}\]

D)
           16/3
View Solution
56)

Eccentricity of the ellipse \[4{{x}^{2}}+{{y}^{2}}-8x+2y+1=0\] is

A)
           \[1/\sqrt{3}\]

B)
           \[\sqrt{3}/2\]

C)
           \[1/2\]

D)
           None of these
View Solution
57)

The equation of an ellipse whose eccentricity is 1/2 and the vertices are (4, 0) and (10, 0) is

A)
           \[3{{x}^{2}}+4{{y}^{2}}-42x+120=0\]

B)
           \[3{{x}^{2}}+4{{y}^{2}}+42x+120=0\]

C)
           \[3{{x}^{2}}+4{{y}^{2}}+42x-120=0\]

D)
           \[3{{x}^{2}}+4{{y}^{2}}-42x-120=0\]
View Solution
58)

The equation of the ellipse whose centre is (2, ?3), one of the foci is (3, ?3) and the corresponding vertex is       (4, ?3) is

A)
           \[\frac{{{(x-2)}^{2}}}{3}+\frac{{{(y+3)}^{2}}}{4}=1\]

B)
           \[\frac{{{(x-2)}^{2}}}{4}+\frac{{{(y+3)}^{2}}}{3}=1\]

C)
           \[\frac{{{x}^{2}}}{3}+\frac{{{y}^{2}}}{4}=1\]

D)
           None of these
View Solution
59)

The equation \[14{{x}^{2}}-4xy+11{{y}^{2}}-44x-58y+71=0\] represents             [BIT Ranchi 1986]

A)
           A circle

B)
           An ellipse

C)
           A hyperbola

D)
           A rectangular hyperbola
View Solution
60)

The centre of the ellipse\[\frac{{{(x+y-2)}^{2}}}{9}+\frac{{{(x-y)}^{2}}}{16}=1\] is [EAMCET 1994]

A)
           (0, 0)

B)
           (1, 1)

C)
           (1, 0)

D)
           (0, 1)
View Solution
61)

The equation of an ellipse whose focus (?1, 1), whose directrix is \[x-y+3=0\] and whose eccentricity is \[\frac{1}{2}\], is given by             [MP PET 1993]

A)
           \[7{{x}^{2}}+2xy+7{{y}^{2}}+10x-10y+7=0\]

B)
           \[7{{x}^{2}}-2xy+7{{y}^{2}}-10x+10y+7=0\]

C)
           \[7{{x}^{2}}-2xy+7{{y}^{2}}-10x-10y-7=0\]

D)
           \[7{{x}^{2}}-2xy+7{{y}^{2}}+10x+10y-7=0\]
View Solution
62)

The foci of the ellipse \[25{{(x+1)}^{2}}+9{{(y+2)}^{2}}=225\] are at [MNR 1991; MP PET 1998; UPSEAT 2000]

A)
           (?1, 2) and (?1, ?6)

B)
           (?1, 2) and (6, 1)

C)
           (1, ?2) and (1, ?6)

D)
           (?1, ?2) and (1, 6)
View Solution
63)

The eccentricity of the ellipse \[9{{x}^{2}}+5{{y}^{2}}-30y=0\], is [MNR 1993; Pb. CET 2004]

A)
           1/3

B)
           2/3

C)
           3/4

D)
           None of these
View Solution
64)

The curve represented by \[x=3(\cos t+\sin t)\], \[y=4(\cos t-\sin t)\] is            [EAMCET 1988; DCE 2000]

A)
           Ellipse

B)
           Parabola

C)
           Hyperbola

D)
           Circle
View Solution
65)

Equation \[x=a\cos \theta ,\ y=b\sin \theta (a>b)\] represent a conic section whose eccentricity e is given by

A)
           \[{{e}^{2}}=\frac{{{a}^{2}}+{{b}^{2}}}{{{a}^{2}}}\]

B)
           \[{{e}^{2}}=\frac{{{a}^{2}}+{{b}^{2}}}{{{b}^{2}}}\]

C)
           \[{{e}^{2}}=\frac{{{a}^{2}}-{{b}^{2}}}{{{a}^{2}}}\]

D)
           \[{{e}^{2}}=\frac{{{a}^{2}}-{{b}^{2}}}{{{b}^{2}}}\]
View Solution
66)

The eccentricity of the ellipse \[4{{x}^{2}}+9{{y}^{2}}+8x+36y+4=0\] is               [MP PET 1996]

A)
           \[\frac{5}{6}\]

B)
           \[\frac{3}{5}\]

C)
           \[\frac{\sqrt{2}}{3}\]

D)
           \[\frac{\sqrt{5}}{3}\]
View Solution
67)

The co-ordinates of the foci of the ellipse \[3{{x}^{2}}+4{{y}^{2}}-12x-8y+4=0\] are

A)
           (1, 2), (3, 4)

B)
           (1, 4), (3, 1)

C)
           (1, 1), (3, 1)

D)
           (2, 3), (5, 4)
View Solution
68)

The eccentricity of the curve represented by the equation \[{{x}^{2}}+2{{y}^{2}}-2x+3y+2=0\] is      [Roorkee 1998]

A)
           0

B)
           1/2

C)
           \[1/\sqrt{2}\]

D)
           \[\sqrt{2}\]
View Solution
69)

For the ellipse \[25{{x}^{2}}+9{{y}^{2}}-150x-90y+225=0\]the eccentricity \[e=\]              [Karnataka CET 2004]

A)
           2/5

B)
           3/5

C)
           4/5

D)
           1/5
View Solution
70)

The eccentricity of the ellipse \[\frac{{{(x-1)}^{2}}}{9}+\frac{{{(y+1)}^{2}}}{25}=1\] is [AMU 1999]

A)
           4/5

B)
           3/5

C)
           5/4

D)
           Imaginary
View Solution
71)

The length of the axes of the conic \[9{{x}^{2}}+4{{y}^{2}}-6x+4y+1=0\], are     [Orissa JEE 2002]

A)
           \[\frac{1}{2},\ 9\]

B)
           \[3,\ \frac{2}{5}\]

C)
           \[1,\ \frac{2}{3}\]

D)
           3, 2
View Solution
72)

The eccentricity of the ellipse \[9{{x}^{2}}+5{{y}^{2}}-18x-2y-16=0\] is               [EAMCET 2003]

A)
           1/2

B)
           2/3

C)
           1/3

D)
           3/4
View Solution
73)

The eccentricity of the conic \[4{{x}^{2}}+16{{y}^{2}}-24x-3y=1\] is [MP PET 2004]

A)
           \[\frac{\sqrt{3}}{2}\]

B)
           \[\frac{1}{2}\]

C)
           \[\frac{\sqrt{3}}{4}\]

D)
           \[\sqrt{3}\]
View Solution
74)

If the line \[y=2x+c\] be a tangent to the ellipse \[\frac{{{x}^{2}}}{8}+\frac{{{y}^{2}}}{4}=1\], then \[c=\]       [MNR 1979; DCE 2000]

A)
           \[\pm 4\]

B)
           \[\pm 6\]

C)
           \[\pm 1\]

D)
           \[\pm 8\]
View Solution
75)

The position of the point (4, ?3) with respect to the ellipse \[2{{x}^{2}}+5{{y}^{2}}=20\] is

A)
           Outside the ellipse

B)
           On the ellipse

C)
           On the major axis

D)
           None of these
View Solution
76)

The equation of the tangent to the ellipse \[{{x}^{2}}+16{{y}^{2}}=16\] making an angle of \[{{60}^{o}}\]with x-axis is

A)
           \[\sqrt{3}x-y+7=0\]

B)
           \[\sqrt{3}x-y-7=0\]

C)
           \[\sqrt{3}x-y\pm 7=0\]

D)
           None of these
View Solution
77)

The position of the point (1, 3) with respect to the ellipse \[4{{x}^{2}}+9{{y}^{2}}-16x-54y+61=0\] [MP PET 1991]

A)
           Outside the ellipse

B)
           On the ellipse

C)
           On the major axis

D)
           On the minor axis
View Solution
78)

The line \[lx+my-n=0\] will be tangent to the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\], if

A)
           \[{{a}^{2}}{{l}^{2}}+{{b}^{2}}{{m}^{2}}={{n}^{2}}\]

B)
           \[a{{l}^{2}}+b{{m}^{2}}={{n}^{2}}\]

C)
           \[{{a}^{2}}l+{{b}^{2}}m=n\]

D)
           None of these
View Solution
79)

The locus of the point of intersection of mutually perpendicular tangent to the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\], is

A)
           A straight line

B)
           A parabola

C)
           A circle

D)
           None of these
View Solution
80)

The equation of the tangent at the point (1/4, 1/4) of the ellipse \[\frac{{{x}^{2}}}{4}+\frac{{{y}^{2}}}{12}=1\] is

A)
           \[3x+y=48\]

B)
           \[3x+y=3\]

C)
           \[3x+y=16\]

D)
           None of these
View Solution
81)

The angle between the pair of tangents drawn to the ellipse \[3{{x}^{2}}+2{{y}^{2}}=5\] from the point (1, 2), is         [MNR 1984]

A)
           \[{{\tan }^{-1}}\left( \frac{12}{5} \right)\]

B)
           \[{{\tan }^{-1}}(6\sqrt{5})\]

C)
           \[{{\tan }^{-1}}\left( \frac{12}{\sqrt{5}} \right)\]

D)
           \[{{\tan }^{-1}}(12\sqrt{5})\]
View Solution
82)

The equations of the tangents of the ellipse \[9{{x}^{2}}+16{{y}^{2}}=144\] which passes through the point (2, 3) is [MP PET 1996]

A)
           \[y=3,\ x+y=5\]

B)
           \[y=-3,\ x-y=5\]

C)
           \[y=4,\ x+y=3\]

D)
           \[y=-4,\ x-y=3\]
View Solution
83)

If any tangent to the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\]cuts off intercepts of length h and k on the axes, then \[\frac{{{a}^{2}}}{{{h}^{2}}}+\frac{{{b}^{2}}}{{{k}^{2}}}=\]

A)
           0

B)
           1

C)
           ?1

D)
           None of these
View Solution
84)

If the line \[y=mx+c\]touches the ellipse \[\frac{{{x}^{2}}}{{{b}^{2}}}+\frac{{{y}^{2}}}{{{a}^{2}}}=1\], then \[c=\]           [MNR 1975; MP PET 1994, 95, 99]

A)
           \[\pm \sqrt{{{b}^{2}}{{m}^{2}}+{{a}^{2}}}\]

B)
           \[\pm \sqrt{{{a}^{2}}{{m}^{2}}+{{b}^{2}}}\]

C)
           \[\pm \sqrt{{{b}^{2}}{{m}^{2}}-{{a}^{2}}}\]

D)
           \[\pm \sqrt{{{a}^{2}}{{m}^{2}}-{{b}^{2}}}\]
View Solution
85)

The ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] and the straight line \[y=mx+c\] intersect in real points only if         [MNR 1995]

A)
           \[{{a}^{2}}{{m}^{2}}<{{c}^{2}}-{{b}^{2}}\]

B)
           \[{{a}^{2}}{{m}^{2}}>{{c}^{2}}-{{b}^{2}}\]

C)
           \[{{a}^{2}}{{m}^{2}}\ge {{c}^{2}}-{{b}^{2}}\]

D)
           \[c\ge b\]
View Solution
86)

If \[y=mx+c\] is tangent on the ellipse \[\frac{{{x}^{2}}}{9}+\frac{{{y}^{2}}}{4}=1\], then the value of c is

A)
           0

B)
           \[3/m\]

C)
           \[\pm \sqrt{9{{m}^{2}}+4}\]

D)
           \[\pm 3\sqrt{1+{{m}^{2}}}\]
View Solution
87)

The locus of the point of intersection of the perpendicular tangents to the ellipse \[\frac{{{x}^{2}}}{9}+\frac{{{y}^{2}}}{4}=1\] is [Karnataka CET 2003]

A)
           \[{{x}^{2}}+{{y}^{2}}=9\]

B)
           \[{{x}^{2}}+{{y}^{2}}=4\]

C)
           \[{{x}^{2}}+{{y}^{2}}=13\]

D)
           \[{{x}^{2}}+{{y}^{2}}=5\]
View Solution
88)

The eccentric angles of the extremities of latus recta of the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] are given by

A)
           \[{{\tan }^{-1}}\left( \pm \frac{ae}{b} \right)\]

B)
           \[{{\tan }^{-1}}\left( \pm \frac{be}{a} \right)\]

C)
           \[{{\tan }^{-1}}\left( \pm \frac{b}{ae} \right)\]

D)
           \[{{\tan }^{-1}}\left( \pm \frac{a}{be} \right)\]
View Solution
89)

Eccentric angle of a point on the ellipse \[{{x}^{2}}+3{{y}^{2}}=6\] at a distance 2 units from the centre of the ellipse is [WB JEE 1990]

A)
           \[\frac{\pi }{4}\]

B)
           \[\frac{\pi }{3}\]

C)
           \[\frac{3\pi }{4}\]

D)
           \[\frac{2\pi }{3}\]
View Solution
90)

The equation of the tangents drawn at the ends of the major axis of the ellipse \[9{{x}^{2}}+5{{y}^{2}}-30y=0\], are [MP PET 1999]

A)
           \[y=\pm 3\]

B)
           \[x=\pm \sqrt{5}\]

C)
           \[y=0,\ y=6\]

D)
           None of these
View Solution
91)

The equation of the normal to the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] at the point \[(a\cos \theta ,\ b\sin \theta )\] is

A)
           \[\frac{ax}{\sin \theta }-\frac{by}{\cos \theta }={{a}^{2}}-{{b}^{2}}\]

B)
           \[\frac{ax}{\sin \theta }-\frac{by}{\cos \theta }={{a}^{2}}+{{b}^{2}}\]

C)
           \[\frac{ax}{\cos \theta }-\frac{by}{\sin \theta }={{a}^{2}}-{{b}^{2}}\]

D)
           \[\frac{ax}{\cos \theta }-\frac{by}{\sin \theta }={{a}^{2}}+{{b}^{2}}\]
View Solution
92)

If the normal at the point \[P(\theta )\] to the ellipse \[\frac{{{x}^{2}}}{14}+\frac{{{y}^{2}}}{5}=1\] intersects it again at the point \[Q(2\theta )\], then \[\cos \theta \] is equal to

A)
           \[\frac{2}{3}\]

B)
           \[-\frac{2}{3}\]

C)
           \[\frac{3}{2}\]

D)
           \[-\frac{3}{2}\]
View Solution
93)

The line \[y=mx+c\]is a normal to the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{a}^{2}}}=1\], if \[c=\]

A)
           \[-(2am+b{{m}^{2}})\]

B)
           \[\frac{({{a}^{2}}+{{b}^{2}})m}{\sqrt{{{a}^{2}}+{{b}^{2}}{{m}^{2}}}}\]

C)
           \[-\frac{({{a}^{2}}-{{b}^{2}})m}{\sqrt{{{a}^{2}}+{{b}^{2}}{{m}^{2}}}}\]

D)
           \[\frac{({{a}^{2}}-{{b}^{2}})m}{\sqrt{{{a}^{2}}+{{b}^{2}}}}\]
View Solution
94)

The equation of normal at the point (0, 3) of the ellipse \[9{{x}^{2}}+5{{y}^{2}}=45\] is    [MP PET 1998]

A)
           \[y-3=0\]

B)
           \[y+3=0\]

C)
           x-axis

D)
           y-axis
View Solution
95)

The equation of the normal at the point (2, 3) on the ellipse \[9{{x}^{2}}+16{{y}^{2}}=180\], is        [MP PET 2000]

A)
           \[3y=8x-10\]

B)
           \[3y-8x+7=0\]

C)
           \[8y+3x+7=0\]

D)
           \[3x+2y+7=0\]
View Solution
96)

If the line \[x\cos \alpha +y\sin \alpha =p\] be normal to the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\], then         [MP PET 2001]

A)
           \[{{p}^{2}}({{a}^{2}}{{\cos }^{2}}\alpha +{{b}^{2}}{{\sin }^{2}}\alpha )={{a}^{2}}-{{b}^{2}}\]

B)
           \[{{p}^{2}}({{a}^{2}}{{\cos }^{2}}\alpha +{{b}^{2}}{{\sin }^{2}}\alpha )={{({{a}^{2}}-{{b}^{2}})}^{2}}\]

C)
           \[{{p}^{2}}({{a}^{2}}{{\sec }^{2}}\alpha +{{b}^{2}}\text{cose}{{\text{c}}^{2}}\alpha )={{a}^{2}}-{{b}^{2}}\]

D)
           \[{{p}^{2}}({{a}^{2}}{{\sec }^{2}}\alpha +{{b}^{2}}\text{cose}{{\text{c}}^{2}}\alpha )={{({{a}^{2}}-{{b}^{2}})}^{2}}\]
View Solution
97)

The line \[lx+my+n=0\]is a normal to the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\], if              [DCE 2000]

A)
           \[\frac{{{a}^{2}}}{{{m}^{2}}}+\frac{{{b}^{2}}}{{{l}^{2}}}=\frac{({{a}^{2}}-{{b}^{2}})}{{{n}^{2}}}\]

B)
           \[\frac{{{a}^{2}}}{{{l}^{2}}}+\frac{{{b}^{2}}}{{{m}^{2}}}=\frac{{{({{a}^{2}}-{{b}^{2}})}^{2}}}{{{n}^{2}}}\]

C)
                \[\frac{{{a}^{2}}}{{{l}^{2}}}-\frac{{{b}^{2}}}{{{m}^{2}}}=\frac{{{({{a}^{2}}-{{b}^{2}})}^{2}}}{{{n}^{2}}}\]

D)
                None of these
View Solution
98)

The equation of tangent and normal at point (3, ?2) of ellipse \[4{{x}^{2}}+9{{y}^{2}}=36\] are       [MP PET 2004]

A)
           \[\frac{x}{3}-\frac{y}{2}=1,\ \frac{x}{2}+\frac{y}{3}=\frac{5}{6}\]

B)
           \[\frac{x}{3}+\frac{y}{2}=1,\ \frac{x}{2}-\frac{y}{3}=\frac{5}{6}\]

C)
           \[\frac{x}{2}+\frac{y}{3}=1,\ \frac{x}{3}-\frac{y}{2}=\frac{5}{6}\]

D)
           None of these
View Solution
99)

The value of \[\lambda \], for which the line \[2x-\frac{8}{3}\lambda y=-3\] is a normal to the conic \[{{x}^{2}}+\frac{{{y}^{2}}}{4}=1\] is [MP PET 2004]

A)
           \[\frac{\sqrt{3}}{2}\]

B)
           \[\frac{1}{2}\]

C)
           \[-\frac{\sqrt{3}}{2}\]

D)
           \[\frac{3}{8}\]
View Solution
100)

The pole of the straight line \[x+4y=4\] with respect to ellipse \[{{x}^{2}}+4{{y}^{2}}=4\] is             [EAMCET 2002]

A)
           (1, 4)

B)
           (1, 1)

C)
           (4, 1)

D)
           (4, 4)
View Solution
101)

In the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\], the equation of diameter conjugate to the diameter \[y=\frac{b}{a}x\], is

A)
           \[y=-\frac{b}{a}x\]

B)
           \[y=-\frac{a}{b}x\]

C)
           \[x=-\frac{b}{a}y\]

D)
           None of these
View Solution
102)

An ellipse has OB as semi minor axis, F and F¢ its foci and the angle FBF¢ is a right angle. Then the eccentricity of the ellipse is [AIEEE 2005]

A)
           \[\frac{1}{4}\]

B)
           \[\frac{1}{\sqrt{3}}\]

C)
           \[\frac{1}{\sqrt{2}}\]

D)
           \[\frac{1}{2}\]
View Solution
103)

If the foci of an ellipse are \[(\pm \sqrt{5},\,0)\] and its eccentricity is \[\frac{\sqrt{5}}{3}\], then the equation of the ellipse is [J & K 2005]

A)
           \[9{{x}^{2}}+4{{y}^{2}}=36\]

B)
           \[4{{x}^{2}}+9{{y}^{2}}=36\]

C)
           \[36{{x}^{2}}+9{{y}^{2}}=4\]

D)
           \[9{{x}^{2}}+36{{y}^{2}}=4\]
View Solution
104)

The sum of the focal distances of any point on the conic \[\frac{{{x}^{2}}}{25}+\frac{{{y}^{2}}}{16}=1\] is                                      [Karnataka CET 2005]

A)
           10

B)
           9

C)
           41

D)
           18
View Solution
105)

Minimum area of the triangle by any tangent to the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] with the coordinate axes is [IIT Screening 2005]

A)
           \[\frac{{{a}^{2}}+{{b}^{2}}}{2}\]

B)
           \[\frac{{{(a+b)}^{2}}}{2}\]

C)
           ab

D)
           \[\frac{{{(a-b)}^{2}}}{2}\]
View Solution
106)

The eccentricity of the ellipse \[25{{x}^{2}}+16{{y}^{2}}-150x-175=0\] is                                                            [Kerala (Engg.) 2005]

A)
           2/5

B)
           2/3

C)
           4/5

D)
           3/4

E)
           3/5
View Solution
107)

The point (4, ?3) with respect to the ellipse \[4{{x}^{2}}+5{{y}^{2}}=1\] [Orissa JEE 2005]

A)
           Lies on the curve

B)
           Is inside the curve

C)
           Is outside the curve

D)
           Is focus of the curve
View Solution
108)

A point ratio of whose distance from a fixed point and line \[x=9/2\] is always 2 : 3. Then locus of the point will be [DCE 2005]

A)
           Hyperbola

B)
           Ellipse

C)
           Parabola

D)
           Circle
View Solution

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JEE Main & Advanced Mathematics Conic Sections Question Bank

done Ellipse Total Questions - 108

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Ellipse Total Questions - 108