If $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2} + \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2} = 1 \left(\mathrm{a} > \mathrm{b}\right)$ and x² - y² = c² cut at right angles, then

• a² + b² = 2c²

• b² - a² = 2c²

• a² - b² = 2c²

• a²b² = 2c²

The equation of the conic with focus at (1, - 1) directrix along x - y + 1 = 0 and with eccentricity $\sqrt{2}$, is

• x² - y² = 1

• xy = 1

• 2xy - 4x + 4y + 1 = 0

• 2xy + 4x - 4y - 1 = 0

The number of common tangents to the circles x² + y² = 4 and x² + y² - 6x - 8y = 24 is

• 0

• 1

• 3

• 4

The locus of the mid-points of the focal chord of the parabola y² = 4ax is

• y² = a(x - a)

• y² = 2a(x - a)

• y² = 4a(x - a)

• None of these

185.

A rod of length l slides with its ends on two perpendicular lines. Then, the locus of its mid point is

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 = \frac{{\mathrm{l}}^2}{4}$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 = \frac{{\mathrm{l}}^2}{2}$

• ${\mathrm{x}}^2 - {\mathrm{y}}^2 = \frac{{\mathrm{l}}^2}{4}$

• None of these

The line joining (5, 0) to ($10\cos \left(\mathrm{\theta }\right), 10\sin \left(\mathrm{\theta }\right)$) is divided internally in the ratio 2 : 3 at P. If 0 varies, then the locus of P is

• a straight line

• a pair of straight lines

• a circle

• None of the above

If 2x + y + k = 0 is a normal to the parabola y² = - 8x, then the value of k, is

• 8

• 16

• 24

• 32

If the equation of an ellipse is 3x² + 2y² + 6x - 8y + 5 = 0, then which of the following are true?

• e = $\frac{1}{\sqrt{3}}$

• centre is (- 1, 2)

• foci are (- 1, 1) are (- 1, 3)

• All of the above

The equation of the common tangents to the two hyperbolas $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2} - \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2} = 1 \mathrm{and} \frac{{\mathrm{y}}^2}{{\mathrm{a}}^2} - \frac{{\mathrm{x}}^2}{{\mathrm{b}}^2} = 1, \mathrm{are}$

• $\mathrm{y} = \pm \mathrm{x} \pm \sqrt{{\mathrm{b}}^2 - {\mathrm{a}}^2}$

• $\mathrm{y} = \pm \mathrm{x} \pm \sqrt{{\mathrm{a}}^2 - {\mathrm{b}}^2}$

• $\mathrm{y} = \pm \mathrm{x} \pm \sqrt{{\mathrm{a}}^2 + {\mathrm{b}}^2}$

• $\mathrm{y} = \pm \mathrm{x} \pm \left({\mathrm{a}}^2 - {\mathrm{b}}^2\right)$

The equation of sphere concentric with the sphere x² + y² + z² - 4x - 6y - 8z - 5 = 0 and which passes through the origin, is

• x² + y² + z² - 4x - 6y - 8z = 0

• x² + y² + z² - 6y - 8z = 0

• x² + y² + z² = 0

• x² + y² + z² - 4x - 6y - 8z - 6 = 0