For the variable , the locus of the point of intersection of the

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81.

For the variable , the locus of the point of intersection of the lines 3tx - 2y + 6t = 0 and 3x + 2ty - 6 = 0 is

  • the ellipse x24 + y29 = 1

  • the ellipse x29 + y24 = 1

  • the hyperbola x24 - y29 = 1

  • the hyperbola x29 - y24 = 1


A.

the ellipse x24 + y29 = 1

Given equation of lines are

3tx - 2y + 6t = 0                 ...(i)

and 3x + 2ty - 6 = 0           ...(ii)

On multiplying Eq. (i) by t and then adding in Eq. (ii), we get

(3t2 + 3)x + 6t2 - 6 = 0

             x = 21 - t21 + t2   x + xt2 = 2 - 2t2 x + 2t2 = 2 - x            t2 = 2 - x2 +x              ...(iii)

On multiplying Eq. (ii) by t and then subtract from Eq. (i), we get

- 2 - 2t2y + 6t + 6t = 0                12t = 21 + t2yOn squaring both sides, we get                 144t2 = 4y21 + t22 1442 - x2 + x = 4y21 + 2 - x2 + x2         from Eq. (iii)   362 - x2 + x = y242 +x2     36 2 - x2 + x = 16y22 + x2   364 - x2 = 16y2      94 - x2= 4y2      36 - 9x2 = 4y2     9x2 + 4y2 = 36

 x24 + y29 = 1, which represents an ellipse.


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82.

The locus of the mid-points of the chords of an ellipse x2 + 4y2 = 4 that are drawn from the positive end of the minor axis, is

  • a circle with centre 12, 0 and radius 1

  • a parabola with focus 12, 0 and directrix x = - 1

  • an ellipse with centre 0, 12, major axis 12 and minor axis

  • a hyperbola with centre 0, 12, transverse axis 1 and conjugate axis 12


83.

A point P lies on the circle x2 + y2 = 169. If Q = (5, 12) and R = (-12, 5) then the QPR is

  • π6

  • π4

  • π3

  • π2


84.

A point moves, so that the sum of squares of its distance from the points (1, 2) and (- 2, 1) is always 6. Then, its locus is

  • the straight line y - 32 = - 3x + 12

  • a circle with centre - 12, 32 and radius 12

  • a parabola with focus (1, 2) and directrix passing through (- 2, 1)

  • an ellipse with foci (1, 2) and (- 2, 1)


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85.

A circle passing through (0, 0), (2, 6), (6, 2) cut the x-axis at the point P  (0, 0). Then, the lenght of OP, where O is the origin, is

  • 52

  • 52

  • 5

  • 10


86.

For the variable t, the locus of the points of intersection of lines x - 2y = t and x + 2y = 1t is

  • the straight line x = y

  • the circle with centre at the origin and radius 1

  • the ellipse with centre at the origin and one focus 25, 0

  • the hyperbola with centre at the origin and one 52, 0


87.

If one end of a diameter of the circle 3x2 + 3y2 - 9x + 6y + y = 0  is (1, 2), then the other end is

  • (2, 1)

  • (2, 4)

  • (2, - 4)

  • (- 4, 2)


88.

The line y = x intersects the hyperbola x29 - y225 = 1 at the points P and Q. The eccentricity of ellipse with PQ as major axis and minor axis of length 52 is

  • 53

  • 53

  • 59

  • 229


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89.

If the distance between the foci of an ellipse is equal to the length of the latusrectum, then its eccentricity is

  • 145 - 1

  • 125 + 1

  • 125 - 1

  • 145 + 1


90.

The equation of the circle passing through the point (1, 1) and the points of intersection of x2 + y2 -  6x - 8 = 0 and x2 + y2 - 6 = 0 is 

  • x2 + y2 + 3x - 5 = 0

  • x2 + y2 - 4x + 2 = 0

  • x2 + y2 + 6x - 4 = 0

  • x2 + y2 - 4y - 2 = 0


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