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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

73.
The value of $λ$ for which the curve (7x + 5)² + (7y + 3)² = $λ^{2}$ (4x + 3y - 24)²represents a parabola is

$\pm \frac{6}{5}$

$\pm \frac{7}{5}$

$\pm \frac{1}{5}$

$\pm \frac{2}{5}$

$\pm \frac{7}{5}$

${(7 x + 5)}^{2} + {(7 y + 3)}^{2} = λ^{2} {(4 x + 3 y - 24)}^{2} \Rightarrow 49 x^{2} + 25 + 70 x + 49 y^{2} + 9 + 42 y = λ^{2} (16 x^{2} + 9 y^{2} + 576 + 24 xy - 144 y - 192 x) \Rightarrow (49 - 16 λ^{2}) x^{2} + (49 - 9 λ^{2}) y^{2} + (70 + 192 λ^{2}) x + (42 + 144 λ^{2}) y - 24 λ^{2} xy + (25 - 576 λ^{2}) = 0 On comparing with {ax}^{2} + 2 hxy + {by}^{2} + 2 gx + 2 fy + c = 0, we get a = 49 - 16 λ^{2}, b = (49 - 9 λ^{2}), h = {(- 12 λ^{2})}^{2} \Rightarrow 2401 - 441 λ^{2} - 784 λ^{2} + 144 λ^{4} = 144 λ^{4} \Rightarrow 2401 - 1225 λ^{2} = 0 \Rightarrow λ^{2} = \frac{2401}{1225} \Rightarrow λ^{2} = \frac{{(49)}^{2}}{{(35)}^{2}} \Rightarrow λ = \pm \frac{49}{35} \Rightarrow λ = \pm \frac{7}{5}$

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