JEE MAIN Hyperbola Previous Year Questions (PYQs) – Page 1 of 3

Consider a hyperbola $\text{H}$ having centre at the origin and foci on the x-axis. Let $C_1$ be the circle touching the hyperbola $\text{H}$ and having the centre at the origin. Let $C_2$ be the circle touching the hyperbola $\text{H}$ at its vertex and having the centre at one of its foci. If areas (in sq units) of $C_1$ and $C_2$ are $36\pi$ and $4\pi$, respectively, then the length (in units) of latus rectum of $\text{H}$ is:

Let the matrix $A=\left[\begin{array}{lll}1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0\end{array}\right]$ satisfy $A^n=A^{n-2}+A^2-I$ for $n \geqslant 3$. Then the sum of all the elements of $\mathrm{A}^{50}$ is :

Let $S = \{(x, y) \in \mathbb{R}^2 : \dfrac{y^2}{1 + r} - \dfrac{x^2}{1 - r} = 1 ; \, r \neq \pm 1 \}$. Then $S$ represents :

If the line $x-1=0$ is a directrix of the hyperbola $kx^{2}-y^{2}=6$, then the hyperbola passes through the point:

If a hyperbola has length of its conjugate axis equal to $5$ and the distance between its foci is $13$, then the eccentricity of the hyperbola is :

If the vertices of a hyperbola are at $(-2,0)$ and $(2,0)$ and one of its foci is at $(-3,0)$, then which one of the following points does not lie on this hyperbola?

Let $H:\dfrac{-x^2}{a^2}+\dfrac{y^2}{b^2}=1$ be the hyperbola whose eccentricity is $\sqrt{3}$ and the length of the latus rectum is $4\sqrt{3}$. Suppose the point $(\alpha,6)$, $\alpha>0$, lies on $H$. If $\beta$ is the product of the focal distances of the point $(\alpha,6)$, then $\alpha^2+\beta$ is equal to:

A hyperbola having the transverse axis of length $\sqrt 2 $ has the same foci as that of the ellipse 3x² + 4y² = 12, then this hyperbola does notpass through which of the following points?

Let $P$ be a point on the hyperbola $H:\ \dfrac{x^2}{9}-\dfrac{y^2}{4}=1$, in the first quadrant, such that the area of the triangle formed by $P$ and the two foci of $H$ is $2\sqrt{13}$. Then, the square of the distance of $P$ from the origin is:

Let $T$ and $C$ respectively be the transverse and conjugate axes of the hyperbola $16x^{2}-y^{2}+64x+4y+44=0$. Then the area of the region above the parabola $x^{2}=y+4$, below the transverse axis $T$ and on the right of the conjugate axis $C$ is: