Hyperbola practice · Easy

If circle ${x^2} + {y^2} + 2\mathrm{g}x + 2fy + k = 0$ intersect hyperbola $xy = {c^2}$ at four points $\left( {{x_i},{y_1}} \right),i = 1,2,3,4$ then

$P F \times PG=$

Let $S=\left\{(x, y) \in R: \frac{y^2}{1+r}-\frac{x^2}{1-r}=1\right\}$, where $r \neq \pm 1$. Then, $S$ represents

The angle between the two asymptotes of the hyperbola $\frac{{{x^2}}}{{16}} - \frac{{{y^2}}}{9} = 1$ is

$P(10,16)$ is a point on the hyperbola having vertices at $(\pm 6,0)$. The equation of the normal to the hyperbola at $P$ is

Let $A$ and $B$ be two fixed points and $P$, another point in the plane, moves in such a way that $k_1 P A+k_2 P B=k_3$, where $k_1, k_2$ and $k_3$ are real constants. The locus of $P$ is
Which one in the options is not true?

The equations of the asymptotes of the hyperbola
$2{x^2} + 5xy + 2{y^2} - 11x - 7y - 4 = 0$ are

The equation of the hyperbola which is referred to its axes as axes of coordinate, and whose distance between the foci is 16 and eccentricity is $\sqrt{2}$, is

The angle between the asymptotes of $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ is equal to

The locus of the point of intersection of the straight lines $\frac{x}{a} + \frac{y}{b} = \lambda$ and $\frac{x}{a} - \frac{y}{b} = \frac{1}{\lambda }$ ( $\lambda$ is a variable) is

A series of hyperbola is drawn having a common transverse axis of length $2 a$. Then the locus of a point $P$ on each hyperbola, such that its distance from the transverse axis is equal to its distance from an asymptote, is

The equation of the hyperbola referred to its axes as coordinate axes, whose latus rectum is 4 and eccentricity is 3 is

If the line $y = 2x + \lambda$ be a tangent to the hyperbola $36{x^2} - 25{y^2} = 3600,$ then $\lambda$ is equal to

If the tangents drawn from a point on the hyperbola $x^2-y^2=a^2-b^2$ to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ make angles $\alpha$ and $\beta$ with the transverse axis of the hyperbola, then

Let $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a > b$ be an ellipse, whose eccentricity is $\frac{1}{\sqrt{2}}$ and the length of the latusrectum is $\sqrt{14}$. Then the square of the eccentricity of $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is:

The eccentricity of the conic represented by $x^2-y^2-4 x+4 y+16=0$, is

The locus of the poles of the chords of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ which subtend a right angle at its centre is

Eccentricity of the hyperbola $2 x^2-2 y^2-3 x y+9 x+12 y=0$ is equal to

The equation of the normal to the hyperbola $\frac{{{x^2}}}{{16}} - \frac{{{y^2}}}{9} = 1$ at $\left( { - 4,0} \right)$ is

For the hyperbola $\frac{{{x^2}}}{{{{\cos }^2}\alpha }} - \frac{{{y^2}}}{{{{\sin }^2}\alpha }} = 1$, which of the following remains constant when $\alpha$ varies?

What is the slope of the tangent line drawn to the hyperbola $x y=a(a \neq 0)$ at the point $(a, 1)$ ?

The locus of a point $\mathrm{P}(\alpha, \beta)$ which is moving under the condition that the line $y=\alpha x+\beta$ is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is

The locus of the point of intersection of the tangents to the hyperbola $\frac{x^2}{16}-\frac{y^2}{36}=1$ which are at right angles is

The centre of the hyperbola $9 x^2-36 x-16 y^2+96 y-252=0$, is

Acceleration of a particle moving along a straight line is a function of velocity as $a=2 \sqrt{v}$. At $t=2 s$, its velocity $v=16 \mathrm{~ms}^{-1}$. Its velocity at $t=3$ s will be

The co-ordinate of focus of $9 x^2-16 y^2+18 x+32 y-151=0$

If the line $l x+m y+n=0$ touches the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, then

The locus of the point of intersection of the tangents at the end-points of normal chords of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, is

Let the foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{7}=1$ and the hyperbola $\frac{x^2}{144}-\frac{y^2}{\alpha}=\frac{1}{25}$ coincide. Then the length of the latus rectum of the hyperbola is:

The coordinates of the centre of the hyperbola ${x^2} + 3xy + 2{y^2} + 2x + 3y + 2 = 0$