JEE Main & Advanced Mathematics Conic Sections Question Bank

done Mock Test - Conic Sections Total Questions - 25

• question_answer1)
  If a circle of constant radius 3k passes through the origin O and meets the coordinate axes at A and B, then the locus of the centroid of triangle OAB is
  A)
  \[{{x}^{2}}+{{y}^{2}}={{(2k)}^{2}}\]         done clear
  B)
  \[{{x}^{2}}+{{y}^{2}}={{(3k)}^{2}}\] done clear
  C)
  \[{{x}^{2}}+{{y}^{2}}={{(4k)}^{2}}\] done clear
  D)
  \[{{x}^{2}}+{{y}^{2}}={{(6k)}^{2}}\] done clear
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• question_answer2)
  If (\[\alpha ,\]\[\beta \]) is a point on the circle whose center is on the x-axis and which touches the line \[x+y=0\]at (2,-2) then the greatest value of \[\alpha \]is
  A)
  \[4-\sqrt{2}\]   done clear
  B)
  6 done clear
  C)
  \[4+2\sqrt{2}\]           done clear
  D)
  \[4+\sqrt{2}\] done clear
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• question_answer3)
  If the tangents are drawn from any point on the line \[x+y=3\]to the circle \[{{x}^{2}}+{{y}^{2}}=9\], then the chord of contact passes through the point
  A)
  (3, 5)    done clear
  B)
  (3, 3) done clear
  C)
  (5, 3)   done clear
  D)
  None of these done clear
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• question_answer4)
  A square is inscribed in the circle \[{{x}^{2}}+{{y}^{2}}-2x+4y-93=0\]with its sides parallel to the coordinate axes. The coordinates of its vertices are
  A)
  (-6, -9), (-6, 5), (8, -9), (8, 5) done clear
  B)
  (-6, 9), (-6, -5), (8, -9), (8, 5) done clear
  C)
  (-6, -9), (-6, 5), (8, 9), (8, 5) done clear
  D)
  (-6, -9), (-6, 5), (8, -9), (8, -5) done clear
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• question_answer5)
  If the chord y=mx+1 of the circle\[{{x}^{2}}+{{y}^{2}}=1\]subtends an angle of measure \[45{}^\circ \]at the major segment of the circle, then the value of m is
  A)
  \[2\pm \sqrt{2}\]         done clear
  B)
  \[-2\pm \sqrt{2}\] done clear
  C)
  \[-1\pm \sqrt{2}\]        done clear
  D)
  None of these done clear
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• question_answer6)
  If the segment intercepted by the parabola \[y=4ax\]with the line \[lx+my+n=0\] subtends a right angle at the vertex, then
  A)
  \[4al+n=0\]   done clear
  B)
  \[4al+4am+n=0\] done clear
  C)
  \[4am+n=0\] done clear
  D)
  \[al+n=0\] done clear
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• question_answer7)
  A circle touches the x-axis and also touches the circle with center (0,3) and radius 2, the locus of center of the circle is
  A)
  A circle            done clear
  B)
  An ellipse done clear
  C)
  A parabola       done clear
  D)
  A hyperbola done clear
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• question_answer8)
  If a line \[y=3x+1\]cuts the parabola \[{{x}^{2}}-4x-4y+20=0\]at A and B, then the tangent of the angle subtended by line segment AB, at the origin is
  A)
  \[8\sqrt{3}/205\]         done clear
  B)
  \[8\sqrt{3}/209\] done clear
  C)
  \[8\sqrt{3}/215\]         done clear
  D)
  None of these done clear
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• question_answer9)
  The vertex of a parabola is the point, (a,b) and the latus rectum is of length, l. the axis of the parabola is parallel to the y-axis and the parabola is concave upward, then its equation is
  A)
   \[{{(x+a)}^{2}}=\frac{1}{2}(2y-2b)\] done clear
  B)
   \[{{(x-a)}^{2}}=\frac{1}{2}(2y-2b)\] done clear
  C)
   \[{{(x+a)}^{2}}=\frac{1}{4}(2y-2b)\] done clear
  D)
   \[{{(x-a)}^{2}}=\frac{1}{8}(2y-2b)\] done clear
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• question_answer10)
  The tangent and normal at the point \[P(a{{t}^{2}},2at)\]to the parabola \[{{y}^{2}}=4ax\]meet the x-axis at T and G, respectively, Then the angle at which the tangent at p to the parabola is inclined to the tangent at p to the circle through P, T and G is
  A)
  \[{{\tan }^{-1}}({{t}^{2}})\]       done clear
  B)
  \[{{\cot }^{-1}}({{t}^{2}})\] done clear
  C)
  \[{{\tan }^{-1}}(t)\] done clear
  D)
  \[{{\cot }^{-1}}(t)\] done clear
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• question_answer11)
  The normal at the point \[(b{{t}_{1}}^{2},2b{{t}_{1}})\]on a parabola meets the parabola again at the point \[(b{{t}_{2}}^{2},2b{{t}_{2}})\]then
  A)
  \[{{t}_{2}}=-{{t}_{1}}-\frac{2}{{{t}_{1}}}\]     done clear
  B)
  \[{{t}_{2}}=-{{t}_{1}}+\frac{2}{{{t}_{1}}}\] done clear
  C)
  \[{{t}_{2}}={{t}_{1}}-\frac{2}{{{t}_{1}}}\]      done clear
  D)
  \[{{t}_{2}}={{t}_{1}}+\frac{2}{{{t}_{1}}}\] done clear
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• question_answer12)
  If the eccentricity of the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}+1}+\frac{{{y}^{2}}}{{{a}^{2}}+2}=1\] Is \[1\sqrt{6}\], then the latus rectum of the ellipse is
  A)
  \[5/\sqrt{6}\]   done clear
  B)
  \[10/\sqrt{6}\] done clear
  C)
  \[8/\sqrt{6}\]   done clear
  D)
  None of these done clear
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• question_answer13)
  Let S and S' be two foci of the ellipse\[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\]if a circle described on SS' as diameter intersects the ellipse at real and distinct points, then the eccentricity e of the ellipse satisfies
  A)
  \[e=1/\sqrt{2}\]          done clear
  B)
  \[e\in (1/\sqrt{2,}1)\] done clear
  C)
  \[e\in (0,1/\sqrt{2,\,})\] done clear
  D)
  None of these done clear
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• question_answer14)
  The liner \[x={{t}^{2}}\]meets the ellipse \[{{x}^{2}}+\frac{{{y}^{2}}}{9}=1\]at real and distinct points if and only if
  A)
  \[\left| t \right|<2\]     done clear
  B)
  \[\left| t \right|<1\] done clear
  C)
  \[\left| t \right|>1\]     done clear
  D)
  None of these done clear
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• question_answer15)
  If the normals at \[P(\theta )\]and \[Q(\pi /2+\theta )\]to the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\]Meet the major axis at G and g, respectively. Then \[P{{G}^{2}}+Q{{g}^{2}}=\]
  A)
  \[{{b}^{2}}(1-{{e}^{2}})(2-{{e}^{2}})\] done clear
  B)
  \[{{a}^{2}}({{e}^{4}}-{{e}^{2}}+2)\] done clear
  C)
  \[{{a}^{2}}(1+{{e}^{2}})(2+{{e}^{2}})\] done clear
  D)
  \[{{b}^{2}}(1+{{e}^{2}})(2+{{e}^{2}})\] done clear
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• question_answer16)
  The ellipse \[{{x}^{2}}+4{{y}^{2}}=4\]is inscribed in a rectangle aligned with the coordinate axes, which is in turn inscribed in another ellipse that passes through the point (4, 0). Then the equation of the ellipse is
  A)
  \[{{x}^{2}}+16{{y}^{2}}=16\] done clear
  B)
  \[{{x}^{2}}+12{{y}^{2}}=16\] done clear
  C)
  \[4{{x}^{2}}+48{{y}^{2}}=48\] done clear
  D)
  \[4{{x}^{2}}+64{{y}^{2}}=48\] done clear
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• question_answer17)
  If the eccentricity of the hyperbola\[{{x}^{2}}-{{y}^{2}}{{\sec }^{2}}\alpha =5\] is \[\sqrt{3}\]times the eccentricity of the ellipse \[{{x}^{2}}{{\sec }^{2}}\alpha +{{y}^{2}}=25\], then a value of \[\alpha \]is
  A)
  \[\pi /6\]           done clear
  B)
  \[\pi /4\] done clear
  C)
  \[\pi /3\]           done clear
  D)
  \[\pi /2\] done clear
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• question_answer18)
  A straight line has its extremities on two fixed straight lines and cuts off from them a triangle of constant area \[{{c}^{2}}\].Then the locus of the middle point of the line is
  A)
  \[2xy={{c}^{2}}\]  done clear
  B)
  \[xy+{{c}^{2}}=0\] done clear
  C)
  \[4{{x}^{2}}{{y}^{2}}=c\]         done clear
  D)
  None of these done clear
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• question_answer19)
  The locus of the feet of the perpendiculars drawn from either focus on a variable tangent to the hyperbola\[16{{y}^{2}}-9{{x}^{2}}=1\]is
  A)
  \[{{x}^{2}}+{{y}^{2}}=9\]       done clear
  B)
  \[{{x}^{2}}+{{y}^{2}}=1/9\] done clear
  C)
  \[{{x}^{2}}+{{y}^{2}}=7/144\] done clear
  D)
  \[{{x}^{2}}+{{y}^{2}}=1/16\] done clear
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• question_answer20)
  The locus of a point P(\[\alpha \],\[\beta \])moving under the condition that the line \[y=\alpha x+\beta \] is a tangent to the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\]is
  A)
  An ellipse         done clear
  B)
  A circle done clear
  C)
  A parabola      done clear
  D)
  A hyperbola done clear
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• question_answer21)
  The circles which can be drawn to pass through (1, 0) and (3, 0) and to touch the y-axis intersect at an angle \[\theta \].the \[cos\theta \]is equal to _________.
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• question_answer22)
  If \[P({{t}^{2}},2t),\,t\in [0,2]\], is an arbitrary point on the parabola \[{{y}^{2}}=4x,Q\]is the foot of perpendicular form focus S on the tangent at p, then the maximum area of \[\Delta PQS\] is
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• question_answer23)
  A man running around a race course notes that the sum of the distances of two fagposts from him is always 10 m and the distance between the flag posts is 8 m then the area of the path he encloses in square meters is _________.
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• question_answer24)
  If the values of m for which the line \[y=mx+2\sqrt{5}\]touch the hyperbola \[16{{x}^{2}}-9{{y}^{2}}=144\]are the roots of the equation \[{{x}^{2}}-(a+b)x-4=0\], then the value of (a + b) is equal to ________. 
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• question_answer25)
  If S=0 is the equation of the hyperbola \[{{x}^{2}}+4xy+3{{y}^{2}}-4x+2y+1=0\], then the value of K for which S+K=0 represents its asymptotes is ________.
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