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Conic Section

Multiple 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

$\frac{213}{\sqrt{17}}$
$\frac{219}{\sqrt{17}}$
$\frac{211}{\sqrt{17}}$
$\frac{210}{\sqrt{17}}$

72.
Let the equation of an ellipse be $\frac{x^{2}}{144} + \frac{y^{2}}{25} = 1$ . Then, the radius of the circle with centre (0, $\sqrt{2}$ ) and passing through the foci of the ellipse is

9

7

11

5

Given equation of ellipse is,

$\frac{x^{2}}{144} + \frac{y^{2}}{25} = 1 Here, a^{2} = 144 and b^{2} = 25 Now, e = \sqrt{1 - \frac{b^{2}}{a^{2}}} = \sqrt{1 - \frac{25}{144}} = \sqrt{\frac{119}{144}} = \frac{\sqrt{119}}{12} ∴ Foci of an ellipse = (\pm ae, 0) = (\pm 12 \times \frac{\sqrt{119}}{12}, 0) = (\pm \sqrt{119}, 0)$

Since, the circle with centre $(0, \sqrt{2})$ and passing through foci ( $\pm \sqrt{119}$ , 0) of the ellipse.

$∴ Radius of circle = \sqrt{{(\sqrt{119} - 0)}^{2} + {(0 - \sqrt{2})}^{2}} = \sqrt{119 + 2} = \sqrt{121} = 11$

73.

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

$\pm \frac{6}{5}$
$\pm \frac{7}{5}$
$\pm \frac{1}{5}$
$\pm \frac{2}{5}$

74.

The equation of the common tangent with positive slope to the parabola y² = 8 $\sqrt{3}$ x and the hyperbola 4x² - y² = 4 is

$y = \sqrt{6} x + \sqrt{2}$
$y = \sqrt{6} x - \sqrt{2}$
$y = \sqrt{3} x + \sqrt{2}$
$y = \sqrt{3} x - \sqrt{2}$

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 $|z - z_{1}| + |z - z_{2}| = 2 |z_{1} - z_{2}|$ 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 $z_{1} \neq \pm 1$ . 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

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