Hyperbola

Let the tangent drawn to the parabola $y^2=24x$ at the point $(\alpha,\beta)$ is perpendicular to the line $2x+2y=5$. Then the normal to the hyperbola $\dfrac{x^2}{\alpha^2}-\dfrac{y^2}{\beta^2}=1$ at the point $(\alpha+4,\beta+4)$ does NOT pass through the point :

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

An ellipse $E:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ passes through the vertices of the hyperbola $H:\dfrac{x^2}{49}-\dfrac{y^2}{64}=-1$. Let the major and minor axes of the ellipse E coincide with the transverse and conjugate axes of the hyperbola H, respectively. Let the product of the eccentricities of E and H be $\dfrac12$. If $l$ is the length of the latus rectum of the ellipse E, then the value of $113l$ is equal to ______.

A common tangent T to the curves $C_1:\dfrac{x^2}4+\dfrac{y^2}9=1$ and $C_2:\dfrac{x^2}{42}-\dfrac{y^2}{143}=1$ does not pass through the fourth quadrant. If T touches $C_1$ at $(x_1,y_1)$ and $C_2$ at $(x_2,y_2)$, then $|2x_1+x_2|$ is equal to ______.

For the hyperbola $H:x^2-y^2=1$ and the ellipse $E:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$, $a>b>0$, let the

(1) eccentricity of E be reciprocal of the eccentricity of H, and

(2) the line $y=\sqrt{\dfrac52}x+K$ be a common tangent of E and H.

Then $4\left(a^2+b^2\right)$ is equal to ______.

Let the hyperbola $H:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ pass through the point $\left(2\sqrt2,-2\sqrt2\right)$. A parabola is drawn whose focus is same as the focus of H with positive abscissa and the directrix of the parabola passes through the other focus of H. If the length of the latus rectum of the parabola is e times the length of the latus rectum of H, where e is the eccentricity of H, then which of the following points lies on the parabola ?

Let the focal chord of the parabola P : $y^2=4x$ along the line L : $y=mx+c$, m > 0 meet the parabola at the points M and N. Let the line L be a tangent to the hyperbola H : $x^2-y^2=4$. If O is the vertex of P and F is the focus of H on the positive $x$-axis, then the area of the quadrilateral OMFN is :

If foci of the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{12}=1$ and the ellipse $\dfrac{x^2}{25}+\dfrac{y^2}9=1$ coincide, then the eccentricity of the hyperbola, with foci $(0,\pm\sqrt{10})$ passing through the point (a, a + 1), is

The value of $\lambda^2$ is ____.

The value of $D$ is ____.

For a complex number $z$, let $\operatorname{Re}(z)$ denote the real part of $z$. Let $S$ be the set of all complex numbers $z$ satisfying $z^4 - |z|^4 = 4iz^2$, where $i = \sqrt{-1}$. Then the minimum possible value of $|z_1 - z_2|^2$, where $z_1, z_2 \in S$ with $\operatorname{Re}(z_1) > 0$ and $\operatorname{Re}(z_2) < 0$, is ______

Let $a$ and $b$ be positive real numbers such that $a > 1$ and $b < a$. Let $P$ be a point in the first quadrant that lies on the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$. Suppose the tangent to the hyperbola at $P$ passes through the point $(1, 0)$, and suppose the normal to the hyperbola at $P$ cuts off equal intercepts on the coordinate axes. Let $\Delta$ denote the area of the triangle formed by the tangent at $P$, the normal at $P$ and the $x$-axis. If $e$ denotes the eccentricity of the hyperbola, then which of the following statements is/are TRUE?

Let $H: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, where $a > b > 0$, be a hyperbola in the $xy$-plane whose conjugate axis $LM$ subtends an angle of $60^\circ$ at one of its vertices $N$. Let the area of the triangle $LMN$ be $4\sqrt{3}$.

The correct option is:

If $2x - y + 1 = 0$ is a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{16} = 1$, then which of the following CANNOT be sides of a right angled triangle?

A hyperbola passes through the point $P\left(\sqrt2,\sqrt3\right)$ and has foci at $(\pm2,0)$. Then the tangent to this hyperbola at P also passes through the point :

The locus of the point of intersection of the straight lines,
$$tx-2y-3t=0$$
$$x-2ty+3=0\quad(t\in\mathbb R),$$
is :

The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is :

Let a and b respectively be the semi-transverse and semi-conjugate axes of a hyperbola whose eccentricity satisfies the equation $9e^2-18e+5=0$. If S(5, 0) is a focus and 5x = 9 is the corresponding directrix of this hyperbola, then $a^2-b^2$ is equal to :

A hyperbola whose transverse axis is along the major axis of the conic, $\dfrac{x^2}3+\dfrac{y^2}4=4$ and has vertices at the foci of this conic. If the eccentricity of the hyperbola is $\dfrac32$, then which of the following points does NOT lie on it ?

Consider the hyperbola $H:x^2-y^2=1$ and a circle $S$ with center $N(x_2,0)$. Suppose that $H$ and $S$ touch each other at a point $P(x_1,y_1)$ with $x_1>1$ and $y_1>0$. The common tangent to $H$ and $S$ at $P$ intersects the $x$-axis at point $M$. If $(l,m)$ is the centroid of the triangle $\Delta PMN$, then the correct expression(s) is(are)