Hyperbola - Coordinate Geometry - Mathematics Previous Year Questions

Let $S$ be the focus of the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ lying on the positive $X$ - axis and $P\left(5, y_1\right)$ be point on the hyperbola. Then $S P=$

If the perimeter of a triangle is 20 and two of its vertices are $(-5,0)$ and $(6,0)$, then the locus of the third vertex is

If $P(\theta)=\left(x_1, \frac{3 \sqrt{5}}{2}\right), 0<\theta<\frac{\pi}{2}$ is a point on the hyperbola $\frac{x^2}{25}-\frac{y^2}{9}=1$, where $\theta$ is the parameter in its parametric form, then $2 x_1+9 \sin ^2 \theta=$

Let $A\left(\theta_1\right)$ and $B\left(\theta_2\right)$ be two points on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $S$ be the focus of the hyperbola, If $A, S, B$ are collinear and

a $\cos \left(\frac{\theta_1+\theta_2}{2}\right)=k \cos \left(\frac{\theta_1-\theta_2}{2}\right)$, then $k=$

If $\frac{x^2}{k-\frac{5}{2}}+\frac{y^2}{\frac{7}{3}-k}=1$ ( $k$ is a real number) represents a hyperbola, then the set of all values of $k$ is

$P(a \sec \theta, b \tan \theta)$ and $Q(a \sec \phi, b \tan \phi)$ are two points on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ where, $\phi+\theta=\frac{\pi}{2}$. If $(h, k)$ is the point of intersection of the normals drawn at $P$ and $Q$, then $k=$

If the equation $x+y+n=0$ represents a normal to the hyperbola $\frac{x^2}{6}-\frac{y^2}{2}=1$, then $n=$

Let $A=(1,2), B=(2,1), C=(-1,-1)$ be three points. If $P$ is a point such that the area of the quadrilateral $P A B C$ is twice the area of the $\triangle P A B$, then the equation of the locus of $P$ is

If $y=m x+4(m>0)$ is a tangent to the hyperbola $\frac{x^2}{25}-\frac{y^2}{9}=1$, then the point of contact of this tangent is

If the line $2 x+\sqrt{6} y=2$ touches the hyperbola $x^2-2 y^2=4$, then the coordinates of the point of contact are

If the angle between the asymptotes of a hyperbola is $30^{\circ}$, then its eccentricity is

If the circle $x^2+y^2=a^2$ intersects the hyperbola $x y=b^2$ at four points $\left(x_1, y_1\right),\left(x_2, y_2\right),\left(x_3, y_3\right),\left(x_4, y_4\right)$, then $y_1 \quad y_2 \quad y_3 y_4=$

The equation of the hyperbola, whose eccentricity is $\sqrt{2}$ and whose foci are 16 units apart, is

A hyperbola having its centre at the origin is passing through the point $(5,2)$ and has transverse axis of length 8 along the $X$-axis. Then, the eccentricity of its conjugate hyperbola is

If $e_1$ is the eccentricity of the hyperbola $x=\sec \theta$, $y=\sqrt{2} \tan \theta$ and $e_2$ is the eccentricity of the hyperbola $x=\sqrt{2} \sec \theta$ and $y=\tan \theta$, then $\frac{e_2^2}{e_1^2}=$

If the latusrectum of a hyperbola subtends an angle of $120^{\circ}$ at its centre, then its eccentricity is

Let $P\left(\frac{\pi}{4}\right), Q\left(\frac{5 \pi}{4}\right), R\left(\frac{3 \pi}{4}\right), T\left(\frac{7 \pi}{4}\right)$ be the points on the hyperbola $x^2-4 y^2-4=0$ in the parametric form. Then the area of the quadrilateral $P Q R T$ is (in square units)

If $P\left(\frac{\pi}{6}\right)$ is a point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1, S, S$ are its foci and $S P+S P=2 | S P-S P$|, then $e=$

Let $S$ be the focus of the hyperbola $x^2-2 y^2=1$ lying on the positive $X$-axis. Let $P(-1,1)$ be a given point. Then, the area of the triangle formed by the line $P S$ with the coordinate axes is (in sq. units)

Let $e_1$ be the eccentricity of a hyperbola for which distance between its focii is 2 times the distance between its directrices and $e_2$ be the eccentricity of another hyperbola for which the length of its transverse axis is twice the length of its conjugate axis. Then, $e_1 e_2=$