Tangent and Normal to a Hyperbola

If the normal to the rectangular hyperbola $x^2-y^2=1$ at the point $P\left(\frac{\pi }{4}\right)$ meets the curve again at $Q\left(\theta \right)$, then ${\sec }^2\theta +\tan \theta =$

If $x$ intercept of tangent drawn to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ on the point $\left(12\sqrt{2},4\right)$ is $6\sqrt{2}$, then $a^2+b^2$ is -

The maximum number of normals to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ from an external point is :

The number of real tangents can be drawn from the point $(16,-3)$ to hyperbola $\frac{x^2}{8}-\frac{y^2}{3}=1$ is

The number of points from where a pair of perpendicular tangents can be drawn to the hyperbola $x^2{\sec }^2\alpha -y^2{\mathrm{cosec}}^2\alpha =1$, $\alpha \in \left(0, \frac{\mathrm{\pi }}{4}\right)$, is

The straight line $x+y=\sqrt{2} p$ will touch the curve $4x^2-9y^2=36$ , if :

If the line $y=mx+7\sqrt{3}$ is normal to the hyperbola $\frac{x^2}{24}-\frac{y^2}{18}=1$, then a value of $m$ is:

If the tangent and the normal to $x^2-y^2=4$ at a point cut off intercepts $a_1, a_2$ on the $x$-axis respectively and $b_1, b_2$ on the $y$-axis respectively then the value of $a_1{a}_2+b_1{b}_2$ is

The equation of the tangent to the hyperbola $x^2-4y^2=36$  which is perpendicular to the line $x-y+4=\mathrm{0,}$ is

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

A hyperbola passes through the point $P\left(\sqrt{2},\sqrt{3}\right)$ and has foci at $\left(\pm 2,0\right)$. Then the tangent to this hyperbola at $P$ also passes through the point

Locus of variable point $P$ which moves in $xy$ plane such that tangents drawn from it to hyperbola $\frac{x^2}{2\sqrt{2}}-\frac{y^2}{2\sqrt{2}}=\sqrt{2}$ are mutually perpendicular, is

Tangent of parabola $x^2=-4ay, a>0$ at one end of latus rectum (say) $P$ is drawn. Point $P$ also lies on hyperbola $\frac{x^2}{{\alpha }^2}-\frac{y^2}{{\beta }^2}=1$, such that tangent at $P$ to hyperbola is parallel to tangent to parabola, then eccentricity of hyperbola lies in the interval

A hyperbola having foci $A\left(4, -1\right)$ and $B\left(4, 5\right)$ has $x+y-7=0$ as one of its tangent, then the point of contact of this tangent is

Common tangents of $9x^2-9y^2=8$ and $y^2=32x$ passes through $(a, 0),$ then $a$ is

If $x+y=k$ is a tangent to the hyperbola $x^2-2y^2=18,$then sum of squares of possible values of $k$ is

Tangents are drawn to the hyperbola $\frac{x^2}{25}-\frac{y^2}{19}=1$, parallel to the line $y=2x$  then point of contact are

The locus of the point of intersection of two tangents of the hyperbola $\frac{x^2}{2}-\frac{y^2}{4}=1,$ if the product of their slopes is $1,$ is

For the hyperbola $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}-\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1,$ distance between the foci is $10$ units. From the point $\left(2,\sqrt{3}\right),$ perpendicular tangents are drawn to the hyperbola, then the value of $\left|\frac{\mathrm{b}}{\mathrm{a}}\right|$ is

Find the value of $m$ for which the line $8y+18=mx$ is tangent to $\frac{x^2}{36}-\frac{y^2}{9}=1$ :