The angle of depressions of the top and the foot of a chimney as seen from the top of a second chimney, which is 150 m high and standing on the same level as the first are θ and ∅ respectively, then the distance between their tops when tan θ = 4/3 and tan ∅ = 5/2 is

• 150/√3 m

• 100√3 m

• 150 m

• 100 m

If (-3, 2) lies on the circle x² + y² + 2gx + 2fy + c = 0, which is concentric with the circle x² + y² + 6x + By - 5 = 0, then c is equal to

• 11

• - 11

• 24

• 100

The eccentricity of the ellipse, which meets the straight line $\frac{\mathrm{x}}{7} + \frac{\mathrm{y}}{2} = 1$ on the axis of x and the staraight line $\frac{\mathrm{x}}{3} - \frac{\mathrm{y}}{5} = 1$ on the axis of y and whose axes lie along the axes of coordinate, is

• $\frac{3\sqrt{2}}{7}$

• $\frac{2\sqrt{6}}{7}$

• $\frac{\sqrt{3}}{7}$

• None of the above

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

378.

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