The equation of the common tangent with positive slope to the par

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 Multiple Choice QuestionsMultiple Choice Questions

71.

If y = 4x + 3 is parallel to a tangent to the parabola y2 = 12x, then its distance from the normal parallel to the given line is

  • 21317

  • 21917

  • 21117

  • 21017


72.

Let the equation of an ellipse be x2144 + y225 = 1. Then, the radius of the circle with centre (0, 2) and passing through the foci of the ellipse is

  • 9

  • 7

  • 11

  • 5


73.

The value of λ for which the curve (7x + 5)2 + (7y + 3)2λ2(4x + 3y - 24)represents a parabola is

  • ± 65

  • ± 75

  • ± 15

  • ± 25


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74.

The equation of the common tangent with positive slope to the parabola y2 = 83x and the hyperbola 4x2 - y2 = 4 is

  • y = 6x + 2

  • y = 6x - 2

  • y = 3x + 2

  • y = 3x - 2


A.

y = 6x + 2

Equation of tangent in slope form of parabola y2 = 83x is,

            y = mx + c            ...(i)where, c = am       c = 23m              ...(ii)

Also, tangent to the hyperbola 4x2 - y2 = 4 is,

c2 = a2m2 - b2

c2 = 1m2 - 4         from Eq. (ii)

 23m2 = m2 - 4             from Eq. (ii)        12m2 = m2 - 4

                  m4 - 4m2 - 12 = 0 m2m2 - 6 + 2m2 - 6 = 0              m2 + 2m2 - 6 = 0 m2 - 6 = 0 and m2 + 2  0 m2 = 6 m = ± 6

i.e. m = 6 is positive slope.

 From Eq. (i),

     y = 6x + 236 y = 6x + 2


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75.

The point on the parabola y2 = 64x which is nearest to the line 4x + 3y + 35 = 0 has coordinates

  • (9, - 24)

  • (1, 81)

  • (4, - 16)

  • (- 9, - 24)


76.

Let z1, z2 be two fixed complex numbers in the argand plane and z be an arbitrary point satisfying z - z1 + z - z2 = 2z1 - z2 Then, the locus of z will be

  • an ellipse

  • a straight line joining z1 and z2

  • a parabola

  • a bisector of the line segment joining z1 and z2


77.

Let z, be a fixed point on the circle of radius 1 centered at the origin in the Argand plane and z1  ± 1. Consider an equilateral triangle inscribed in the circle with z1, z2, z3 as the vertices taken in the counterclockwise direction. Then, z1z2z3 is equal to

  • z12

  • z13

  • z14

  • z1


78.

The equation of hyperbola whose coordinates of the foci are (± 8, 0) and the length of latusrectum is 24 units, is

  • 3x2 - y2 = 48

  • 4x2 - y2 = 48

  • x2 - 3y2 = 48

  • x2 - 4y2 = 48


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79.

If the circle x2 + y2 + 2gx + 2fy + c = 0 cuts the three circles x2 + y2 - 5 = 0, x2 + y2 - 8x - 6y + 10 = 0 and x2 + y2 - 4x + 2y - 2 = 0 at the extremities of  their diameters, then

  • c = - 5

  • fg = 147/25

  • g + 2f = c + 2

  • 4f = 3g


80.

Lines x + y = 1 and 3y = x + 3 intersect the ellipse x2 + 9y2 = 9 at the points P,Q and R. The area of the PQR is

  • 365

  • 185

  • 95

  • 15


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