19,843 questions & answers

Choose the correct alternatives.

  • (i) Acceleration due to gravity increases/decreases with increasing altitude.

  • (ii) Acceleration due to gravity increases/decreases with increasing depth (assume the Earth to be a sphere of uniform density).

  • (iii) Acceleration due to gravity is independent of the mass of the Earth/mass of the body.

  • (iv) The formula $-G M m\left(\frac{1}{r_{2}}-\frac{1}{r_{1}}\right)$ is more/less accurate than the formula $\mathrm{mg}\left(r_{2}-r_{1}\right)$ for the difference of potential energy between two points $r_{2}$ and $r_{1}$ distance away from the centre of the Earth.

If (5,12),(24,7) are the foci of the hyperbola passing through the origin, then its eccentricity is:

A. $ \frac{13}{5} $

B. $ \frac{\sqrt{386}}{13} $

C. $\frac{\sqrt{386}}{25}$

D. $\frac{\sqrt{386}}{12}$

The eccentricity of $\frac{x^{2}}{9} - \frac{y^{2}}{16} = 1$ is:

A $\frac{17}{16}$

B $\frac{5}{4}$

C $\frac{5}{3}$

D $\frac{\sqrt{7}}{4}$

If the foci of the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$ and the hyperbola $\frac{x^{2}}{4}-\frac{y^{2}}{b^{2}}=1$ coincide, then $b^{2}=$

a) 4

b) 5

c) 8

d) 9

If a hyperbola passes through a focus of the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$ and its transverse and conjugate axes coincide with major and minor axes of the ellipse, and the product of their eccentricities is 1, then:

A $\frac{x^{2}}{16}+\frac{y^{2}}{25}=1$ B $\frac{x^{2}}{16}-\frac{y^{2}}{9}=-1$ C $\frac{x^{2}}{9}-\frac{y^{2}}{25}=1$ D $\frac{x^{2}}{16}-\frac{y^{2}}{9}=1"

The equation $\frac{x^{2}}{7-K} + \frac{y^{2}}{5-K} = 1$ represents a hyperbola if:

  • $5 < K < 7$
  • $K < 5$ or $K > 7$
  • $K > 5$
  • $K \neq 5$, $K \neq 7$

The transverse axis of a hyperbola is of length $2a$ and a vertex divides the segment of the axis between the centre and the corresponding focus in the ratio $2:1$, then the equation of the hyperbola is

A. $5x^2 - 4y^2 = 5a^2$

B. $5x^2 - 4y^2 = 4a^2$

C. $4x^2 - 5y^2 = 5a^2$

D. $4x^2 - 5y^2 = 4a^2$

The angle between the asymptotes of the hyperbola $x^{2} / a^{2}-y^{2} / b^{2}=1$ is A $2\sin^{-1}(e)$ B $2\cos^{-1}(e)$ C $2\tan^{-1}(e)$ D $2\sec^{-1}(e)$

The vertices of a hyperbola are $(2,0),(-2,0)$ and the foci are $(3,0),(-3,0)$. The equation of the hyperbola is

A $\frac{x^{2}}{5}-\frac{y^{2}}{4}=1$ B $\frac{x^{2}}{4}-\frac{y^{2}}{5}=1$ C $\frac{x^{2}}{5}-\frac{y^{2}}{2}=1$ D $\frac{x^{2}}{2}-\frac{y^{2}}{5}=1$

The equation of the hyperbola with its transverse axis parallel to the $x$-axis and its center at $(-2,1)$, with a length of the transverse axis of 10 and eccentricity of $6/5$ is:

$\frac{(x+2)^{2}}{25}-\frac{(y-1)^{2}}{11}=1$

Therefore, the correct option is B.

The angle between the asymptotes of a hyperbola given by $x^{2}-3y^{2}=1$ is:

a) $15^{\circ}$ b) $45^{\circ}$ c) $60^{\circ}$ d) $30^{\circ}$

The product of lengths of perpendicular from any point on the hyperbola $x^{2}-y^{2}=16$ to its asymptotes is:
A) 2
B) 4
C) 8
D) 16

Angle between the asymptotes of a hyperbola is $30^\circ$ then $\mathrm{e}=$ A $\sqrt{6}$ B $\sqrt{2}$ C $\sqrt{6}-\sqrt{2}$ D $\sqrt{6}-\sqrt{3}$

Find the eccentricity of the hyperbola with asymptotes $3x + 4y = 2$ and $4x - 3y = 2$.

The angle between the asymptotes of the hyperbola $xy=a^{2}$ is:

a) $30^{\circ}$

b) $60^{\circ}$

c) $45^{\circ}$

d) $90^{\circ}$

The equation to the pair of asymptotes of the hyperbola $\frac{x^{2}}{9}-\frac{y^{2}}{5}=1$ is:

9x^2 - 5y^2 = 0

Equation of normal to $9x^{2}-25y^{2}=225$ at $\theta=\pi / 4$ is

A $5x+3\sqrt{2}y=34\sqrt{2}$ B $5x+\sqrt{2}y=34\sqrt{2}$

C $5x+\sqrt{3}y=34\sqrt{2}$ D $5x-3\sqrt{2}y=34\sqrt{2}$

Find the angle between the asymptotes of the hyperbola $\frac{x^{2}}{16}-\frac{y^{2}}{9}=1`.

The correct answer is: C $\frac{\pi}{6}$

Find the equation of the normal to the hyperbola $x^2 - 3y^2 = 144$ at the positive end of the latus rectum.

The correct answer is: $3x - \sqrt{3}y = 48$

Equation of the tangent to the hyperbola $4x^{2}-9y^{2}=1$ with eccentric angle $\pi / 6$ is

A. $4x+3y=\sqrt{3}$

B. $4x-3y=\sqrt{3}$

C. $3x-4y=\sqrt{3}$

D. $3x-4y=\sqrt{5}$