JEE Main & Advanced Mathematics Circle and System of Circles Question Bank

done Chord of contact of tangent, Pole and Polar Total Questions - 41

• question_answer1)
  The common chord of the circle \[{{x}^{2}}+{{y}^{2}}+4x+1=0\] and \[{{x}^{2}}+{{y}^{2}}+6x+2y+3=0\] is                                                [MP PET 1991]
  A)
             \[x+y+1=0\]                            done clear
  B)
             \[5x+y+2=0\] done clear
  C)
             \[2x+2y+5=0\]                       done clear
  D)
             \[3x+y+3=0\] done clear
  View Solution play_arrow

• question_answer2)
  If the middle point of a chord of the circle \[{{x}^{2}}+{{y}^{2}}+x-y-1=0\]be (1, 1), then  the length of the chord is 
  A)
             4     done clear
  B)
             2 done clear
  C)
             5     done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer3)
  \[y=mx\] is a chord of a circle of radius a and the diameter of the circle lies along x-axis and one end of this chord in origin .The equation of the circle described on this chord as diameter is                                                                      [MP PET 1990]
  A)
             \[(1+{{m}^{2}})({{x}^{2}}+{{y}^{2}})-2ax=0\] done clear
  B)
             \[(1+{{m}^{2}})({{x}^{2}}+{{y}^{2}})-2a(x+my)=0\] done clear
  C)
             \[(1+{{m}^{2}})({{x}^{2}}+{{y}^{2}})+2a(x+my)=0\] done clear
  D)
             \[(1+{{m}^{2}})({{x}^{2}}+{{y}^{2}})-2a(x-my)=0\] done clear
  View Solution play_arrow

• question_answer4)
  The locus of the middle points of those chords of the circle \[{{x}^{2}}+{{y}^{2}}=4\]which subtend a right angle at the origin is  [MP PET 1990; IIT 1984; RPET 1997; DCE 2000, 01]
  A)
             \[{{x}^{2}}+{{y}^{2}}-2x-2y=0\]                                     done clear
  B)
             \[{{x}^{2}}+{{y}^{2}}=4\] done clear
  C)
             \[{{x}^{2}}+{{y}^{2}}=2\]    done clear
  D)
             \[{{(x-1)}^{2}}+{{(y-2)}^{2}}=5\] done clear
  View Solution play_arrow

• question_answer5)
   The equation of the chord of the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\]having \[({{x}_{1}},{{y}_{1}})\]as its mid-point is  [IIT 1983; MP PET 1986; Pb. CET 2003]
  A)
             \[x{{y}_{1}}+y{{x}_{1}}={{a}^{2}}\]                              done clear
  B)
             \[{{x}_{1}}+{{y}_{1}}=a\] done clear
  C)
             \[x{{x}_{1}}+y{{y}_{1}}=x_{1}^{2}+y_{1}^{2}\]      done clear
  D)
             \[x{{x}_{1}}+y{{y}_{1}}={{a}^{2}}\] done clear
  View Solution play_arrow

• question_answer6)
   Locus of the middle points of the chords of the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\]which are parallel to \[y=2x\] will be
  A)
  A circle with radius a                done clear
  B)
  A straight line with slope \[-\frac{1}{2}\] done clear
  C)
  A circle will centre (0, 0)          done clear
  D)
  A straight line with slope - 2 done clear
  View Solution play_arrow

• question_answer7)
  The length of the chord intercepted by the circle \[{{x}^{2}}+{{y}^{2}}={{r}^{2}}\]on the line \[\frac{x}{a}+\frac{y}{b}=1\] is
  A)
             \[\sqrt{\frac{{{r}^{2}}({{a}^{2}}+{{b}^{2}})-{{a}^{2}}{{b}^{2}}}{{{a}^{2}}+{{b}^{2}}}}\]            done clear
  B)
             \[2\sqrt{\frac{{{r}^{2}}({{a}^{2}}+{{b}^{2}})-{{a}^{2}}{{b}^{2}}}{{{a}^{2}}+{{b}^{2}}}}\] done clear
  C)
             \[2\frac{\sqrt{{{r}^{2}}({{a}^{2}}+{{b}^{2}})-{{a}^{2}}{{b}^{2}}}}{{{a}^{2}}+{{b}^{2}}}\]          done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer8)
  Middle point of the chord of the circle \[{{x}^{2}}+{{y}^{2}}=25\] intercepted on the line \[x-2y=2\]is
  A)
             \[\left( \frac{3}{5},\frac{4}{5} \right)\]                                      done clear
  B)
             \[(-2,-2)\] done clear
  C)
             \[\left( \frac{2}{5},-\frac{4}{5} \right)\]                                     done clear
  D)
             \[\left( \frac{8}{3},\frac{1}{3} \right)\] done clear
  View Solution play_arrow

• question_answer9)
  If the line \[x-2y=k\]cuts off a chord of length 2 from the circle \[{{x}^{2}}+{{y}^{2}}=3\], then k =
  A)
             0     done clear
  B)
             \[\pm 1\] done clear
  C)
             \[\pm \sqrt{10}\]                       done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer10)
  From the origin chords are drawn to the circle\[{{(x-1)}^{2}}+{{y}^{2}}=1\]. The equation of the locus of the middle points of these chords is [IIT 1985; EAMCET 1991]
  A)
             \[{{x}^{2}}+{{y}^{2}}-3x=0\]                                           done clear
  B)
             \[{{x}^{2}}+{{y}^{2}}-3y=0\] done clear
  C)
             \[{{x}^{2}}+{{y}^{2}}-x=0\] done clear
  D)
             \[{{x}^{2}}+{{y}^{2}}-y=0\] done clear
  View Solution play_arrow

• question_answer11)
  The polars drawn  from (-1, 2) to the circles \[{{S}_{1}}\equiv {{x}^{2}}+{{y}^{2}}+6y+7=0\]and \[{{S}_{2}}\equiv {{x}^{2}}+{{y}^{2}}+6x+1=0\], are  [RPET 2002]
  A)
  Parallel                                      done clear
  B)
  Equal done clear
  C)
  Perpendicular                           done clear
  D)
  Intersect at a point done clear
  View Solution play_arrow

• question_answer12)
  The equation of the diameter of the circle \[{{x}^{2}}+{{y}^{2}}+2x-4y-11=0\]which bisects the chords intercepted on the line \[2x-y+3=0\]is
  A)
             \[x+y-7=0\]                              done clear
  B)
             \[2x-y-5=0\] done clear
  C)
             \[x+2y-3=0\]                           done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer13)
  If the lengths of the chords intercepted by the circle \[{{x}^{2}}+{{y}^{2}}+2gx+2fy=0\]from the co-ordinate axes be 10 and 24 respectively, then the radius of the circle is
  A)
             17   done clear
  B)
             9 done clear
  C)
             14   done clear
  D)
             13 done clear
  View Solution play_arrow

• question_answer14)
  The equation of the common chord of the circles \[{{(x-a)}^{2}}+{{(y-b)}^{2}}={{c}^{2}}\]and \[{{(x-b)}^{2}}+{{(y-a)}^{2}}={{c}^{2}}\] is
  A)
             \[x-y=0\]                                   done clear
  B)
             \[x+y=0\] done clear
  C)
             \[x+y={{a}^{2}}+{{b}^{2}}\]                                           done clear
  D)
             \[x-y={{a}^{2}}-{{b}^{2}}\] done clear
  View Solution play_arrow

• question_answer15)
  The length of common chord of the circles \[{{(x-a)}^{2}}+{{y}^{2}}={{a}^{2}}\]and \[{{x}^{2}}+{{(y-b)}^{2}}={{b}^{2}}\]is                    [MP PET 1989]
  A)
             \[2\sqrt{{{a}^{2}}+{{b}^{2}}}\]                                        done clear
  B)
             \[\frac{ab}{\sqrt{{{a}^{2}}+{{b}^{2}}}}\] done clear
  C)
             \[\frac{2ab}{\sqrt{{{a}^{2}}+{{b}^{2}}}}\]                     done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer16)
  The co-ordinates of pole of line \[lx+my+n=0\]with respect to circles \[{{x}^{2}}+{{y}^{2}}=1\], is                                              [RPET 1987]
  A)
             \[\left( \frac{l}{n},\frac{m}{n} \right)\]                                       done clear
  B)
             \[\left( -\frac{l}{n},-\frac{m}{n} \right)\] done clear
  C)
             \[\left( \frac{l}{n},-\frac{m}{n} \right)\]                                     done clear
  D)
             \[\left( -\frac{l}{n},\frac{m}{n} \right)\] done clear
  View Solution play_arrow

• question_answer17)
  The length of common chord of the circles \[{{x}^{2}}+{{y}^{2}}=12\]and \[{{x}^{2}}+{{y}^{2}}-4x+3y-2=0\], is                              [RPET 1990, 99]
  A)
             \[4\sqrt{2}\]                              done clear
  B)
             \[5\sqrt{2}\] done clear
  C)
             \[2\sqrt{2}\]                              done clear
  D)
             \[6\sqrt{2}\] done clear
  View Solution play_arrow

• question_answer18)
  The length of the common chord of the circles \[{{(x-a)}^{2}}+{{(y-b)}^{2}}={{c}^{2}}\]and \[{{(x-b)}^{2}}+{{(y-a)}^{2}}={{c}^{2}}\], is
  A)
             \[\sqrt{4{{c}^{2}}-2{{(a-b)}^{2}}}\]                                  done clear
  B)
             \[\sqrt{4{{c}^{2}}+2{{(a-b)}^{2}}}\] done clear
  C)
             \[\sqrt{4{{c}^{2}}-2{{(a+b)}^{2}}}\]                                done clear
  D)
             \[\sqrt{4{{c}^{2}}+2{{(a+b)}^{2}}}\] done clear
  View Solution play_arrow

• question_answer19)
  The locus of the middle points of chords of the circle \[{{x}^{2}}+{{y}^{2}}-2x-6y-10=0\] which passes through the origin, is [Roorkee 1989]
  A)
             \[{{x}^{2}}+{{y}^{2}}+x+3y=0\]                                   done clear
  B)
             \[{{x}^{2}}+{{y}^{2}}-x+3y=0\] done clear
  C)
             \[{{x}^{2}}+{{y}^{2}}+x-3y=0\]                                     done clear
  D)
             \[{{x}^{2}}+{{y}^{2}}-x-3y=0\] done clear
  View Solution play_arrow

• question_answer20)
   The distance between the chords of contact of the tangents to the circle \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\]from the origin and the point \[(g,f)\]is
  A)
             \[\frac{1}{2}\left( \frac{{{g}^{2}}+{{f}^{2}}-c}{\sqrt{{{g}^{2}}+{{f}^{2}}}} \right)\]      done clear
  B)
              \[\left( \frac{{{g}^{2}}+{{f}^{2}}-c}{\sqrt{{{g}^{2}}+{{f}^{2}}}} \right)\] done clear
  C)
             \[\frac{1}{2}\left( \frac{{{g}^{2}}+{{f}^{2}}-c}{{{g}^{2}}+{{f}^{2}}} \right)\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer21)
  The pole of the straight line \[x+2y=1\]with respect to the circle \[{{x}^{2}}+{{y}^{2}}=5\]is                                          [RPET 2000, 01]
  A)
             (5, 5)                                          done clear
  B)
           (5, 10) done clear
  C)
             (10, 5)                                       done clear
  D)
             (10, 10) done clear
  View Solution play_arrow

• question_answer22)
  A line \[lx+my+n=0\]meets the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\]at the points P and Q. The tangents drawn at the points P and Q meet at R, then the coordinates of R is
  A)
             \[\left( \frac{{{a}^{2}}l}{n},\frac{{{a}^{2}}m}{n} \right)\]            done clear
  B)
             \[\left( \frac{-{{a}^{2}}l}{n},\frac{-{{a}^{2}}m}{n} \right)\] done clear
  C)
             \[\left( \frac{{{a}^{2}}n}{l},\frac{{{a}^{2}}n}{m} \right)\]            done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer23)
  Polar of origin (0, 0) with respect to the circle \[{{x}^{2}}+{{y}^{2}}+2\lambda x+2\mu y+c=0\] touches the circle \[{{x}^{2}}+{{y}^{2}}={{r}^{2}}\], if  [RPET 1992]
  A)
             \[c=r({{\lambda }^{2}}+{{\mu }^{2}})\]                          done clear
  B)
             \[r=c\,({{\lambda }^{2}}+{{\mu }^{2}})\] done clear
  C)
             \[{{c}^{2}}={{r}^{2}}({{\lambda }^{2}}+{{\mu }^{2}})\]    done clear
  D)
             \[{{r}^{2}}={{c}^{2}}({{\lambda }^{2}}+{{\mu }^{2}})\] done clear
  View Solution play_arrow

• question_answer24)
  Tangents AB and AC are drawn from the point \[A(0,\,1)\]to the circle \[{{x}^{2}}+{{y}^{2}}-2x+4y+1=0\]. Equation of the circle through A, B and C is
  A)
             \[{{x}^{2}}+{{y}^{2}}+x+y-2=0\]                                  done clear
  B)
             \[{{x}^{2}}+{{y}^{2}}-x+y-2=0\] done clear
  C)
             \[{{x}^{2}}+{{y}^{2}}+x-y-2=0\]                                    done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer25)
  Length of the common chord of the circles \[{{x}^{2}}+{{y}^{2}}+5x+7y+9=0\]and \[{{x}^{2}}+{{y}^{2}}+7x+5y+9=0\]is  [Kurukshetra CEE 1996]
  A)
             9     done clear
  B)
             8 done clear
  C)
             7     done clear
  D)
             6 done clear
  View Solution play_arrow

• question_answer26)
  The locus of midpoint of the chords of the circle \[{{x}^{2}}+{{y}^{2}}-2x-2y-2=0\]which makes an angle of \[120{}^\circ \] at the centre is [MNR 1994]
  A)
             \[{{x}^{2}}+{{y}^{2}}-2x-2y+1=0\]                               done clear
  B)
             \[{{x}^{2}}+{{y}^{2}}+x+y-1=0\] done clear
  C)
             \[{{x}^{2}}+{{y}^{2}}-2x-2y-1=0\]                                 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer27)
  If the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\]cuts off a chord of length 2b from the line \[y=mx+c\], then 
  A)
             \[(1-{{m}^{2}})({{a}^{2}}+{{b}^{2}})={{c}^{2}}\]              done clear
  B)
             \[(1+{{m}^{2}})({{a}^{2}}-{{b}^{2}})={{c}^{2}}\] done clear
  C)
             \[(1-{{m}^{2}})({{a}^{2}}-{{b}^{2}})={{c}^{2}}\]  done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer28)
   The pole of the straight line \[9x+y-28=0\]with respect to circle \[2{{x}^{2}}+2{{y}^{2}}-3x+5y-7=0\], is [RPET 1990, 99; MNR 1984; UPSEAT  2000]
  A)
             (3, 1)                                          done clear
  B)
             (1, 3) done clear
  C)
             (3, -1)                                        done clear
  D)
             (-3, 1) done clear
  View Solution play_arrow

• question_answer29)
  If polar of a circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\]with respect to \[(x',y')\] is \[Ax+By+C=0\], then its pole will be             [RPET 1995]
  A)
             \[\left( \frac{{{a}^{2}}A}{-C},\frac{{{a}^{2}}B}{-C} \right)\]        done clear
  B)
             \[\left( \frac{{{a}^{2}}A}{C},\frac{{{a}^{2}}B}{C} \right)\] done clear
  C)
             \[\left( \frac{{{a}^{2}}C}{A},\frac{{{a}^{2}}C}{B} \right)\]          done clear
  D)
             \[\left( \frac{{{a}^{2}}C}{-A},\frac{{{a}^{2}}C}{-B} \right)\] done clear
  View Solution play_arrow

• question_answer30)
  The polar of the point (5, ?1/2) w.r.t  circle \[{{(x-2)}^{2}}+{{y}^{2}}=4\]is                                        [RPET 1996]
  A)
             \[5x-10y+2=0\]                       done clear
  B)
             \[6x-y-20=0\] done clear
  C)
             \[10x-y-10=0\]                         done clear
  D)
             \[x-10y-2=0\] done clear
  View Solution play_arrow

• question_answer31)
  The pole of the line \[2x+3y=4\]w.r.t circle \[{{x}^{2}}+{{y}^{2}}=64\] is  [RPET 1996]
  A)
             (32, 48)                                     done clear
  B)
             (48, 32) done clear
  C)
             (- 32, 48)                                   done clear
  D)
             (48, -32) done clear
  View Solution play_arrow

• question_answer32)
  The length  of the common chord of the circles \[{{x}^{2}}+{{y}^{2}}+2x+3y+1=0\]and \[{{x}^{2}}+{{y}^{2}}+4x+3y+2=0\]is  [MP PET 2000]
  A)
             \[9/2\]                                         done clear
  B)
             \[2\sqrt{2}\] done clear
  C)
             \[3\sqrt{2}\]                              done clear
  D)
             \[3/2\] done clear
  View Solution play_arrow

• question_answer33)
  The length of the chord joining the points in which the straight line \[\frac{x}{3}+\frac{y}{4}=1\]cuts the circle \[{{x}^{2}}+{{y}^{2}}=\frac{169}{25}\]is  [Orissa JEE 2003]
  A)
             1     done clear
  B)
             2 done clear
  C)
             4     done clear
  D)
             8 done clear
  View Solution play_arrow

• question_answer34)
  Which of the following is a point on the common chord of the circles \[{{x}^{2}}+{{y}^{2}}+2x-3y+6=0\]and \[{{x}^{2}}+{{y}^{2}}+x-8y-13=0\] Karnataka CET 2003]
  A)
             (1, -2)                                        done clear
  B)
             (1, 4) done clear
  C)
             (1, 2)                                          done clear
  D)
             (1, -4) done clear
  View Solution play_arrow

• question_answer35)
  If the circle \[{{x}^{2}}+{{y}^{2}}=4\]bisects the circumference of the circle \[{{x}^{2}}+{{y}^{2}}-2x+6y+a=0\], then a equals                    [RPET 1999]
  A)
             4     done clear
  B)
             -4 done clear
  C)
             16   done clear
  D)
             -16 done clear
  View Solution play_arrow

• question_answer36)
  The equation of polar of the point (1, 2)with respect to the circle \[{{x}^{2}}+{{y}^{2}}=7\], is       [RPET 1983, 84; MNR 1973]
  A)
             \[x-2y-7=0\]                             done clear
  B)
             \[x+2y-7=0\] done clear
  C)
             \[x-2y=0\]                                 done clear
  D)
             \[x+2y=0\] done clear
  View Solution play_arrow

• question_answer37)
  The equation of the chord of contact, if the tangents are drawn from the point (5, ?3) to the circle \[{{x}^{2}}+{{y}^{2}}=10\], is
  A)
             \[5x-3y=10\]                            done clear
  B)
             \[5x+3y=10\] done clear
  C)
             \[3x+5y=10\]                          done clear
  D)
             \[3x-5y=10\] done clear
  View Solution play_arrow

• question_answer38)
  A line through (0,0) cuts the circle \[{{x}^{2}}+{{y}^{2}}-2ax=0\] at A and B, then locus of the centre of the circle drawn on AB as a diameter is [RPET 2002]
  A)
             \[{{x}^{2}}+{{y}^{2}}-2ay=0\]                                         done clear
  B)
             \[{{x}^{2}}+{{y}^{2}}+ay=0\] done clear
  C)
             \[{{x}^{2}}+{{y}^{2}}+ax=0\]                                         done clear
  D)
             \[{{x}^{2}}+{{y}^{2}}-ax=0\] done clear
  View Solution play_arrow

• question_answer39)
  The radius of the circle, having centre at (2,1) whose one of the chord is a diameter of the circle \[{{x}^{2}}+{{y}^{2}}-2x-6y+6=0\] is  [IIT Screening 2004]
  A)
             1     done clear
  B)
             2 done clear
  C)
             3     done clear
  D)
             \[\sqrt{3}\] done clear
  View Solution play_arrow

• question_answer40)
  The intercept on the line \[y=x\] by the circle \[{{x}^{2}}+{{y}^{2}}-2x=0\] is AB, equation of the circle on AB as a diameter is [AIEEE 2004]
  A)
             \[{{x}^{2}}+{{y}^{2}}+x-y=0\]                                        done clear
  B)
             \[{{x}^{2}}+{{y}^{2}}-x+y=0\] done clear
  C)
             \[{{x}^{2}}+{{y}^{2}}+x+y=\]0                                      done clear
  D)
           \[{{x}^{2}}+{{y}^{2}}-x-y=0\] done clear
  View Solution play_arrow

• question_answer41)
  The equation of the circle having as a diameter, the chord \[x-y-1=0\] of the circle \[2{{x}^{2}}+2{{y}^{2}}-2x-6y-25=0\], is
  A)
             \[{{x}^{2}}+{{y}^{2}}-3x-y-\frac{29}{2}=0\]                done clear
  B)
             \[2{{x}^{2}}+2{{y}^{2}}+2x-5y-\frac{29}{2}=0\] done clear
  C)
             \[2{{x}^{2}}+2{{y}^{2}}-6x-2y-21=0\] done clear
  D)
             None of these done clear
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