Equations of Tangents in Different Forms

A straight line touches the rectangular hyperbola $9x^2-9y^2=8$ and the parabola $y^2=32x$. The equation of the line is:

The locus of the point of intersection of two perpendicular tangents to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is:

Tangents are drawn from a point on the circle $x^2+y^2=11$ to the hyperbola $\frac{x^2}{36}-\frac{y^2}{25}=1$, then tangents are at angle:

A line through the origin meets the circle $x^2+y^2=a^2$ at $P$ and the hyperbola $x^2-y^2=a^2$ at $Q$. Prove that the locus of the point of intersection of tangent at $P$ to the circle with the tangent at $Q$ to the hyperbola is the curve.

A ray emanating from the point $(-\sqrt{41},0)$ is incident on the hyperbola $16x^2-25y^2=400$ at the point $P$ with abscissa $10$ . Then the equation of the reflected ray after first reflection and point $P$ lies in second quadrant is :

Tangents drawn from a point on the circle $x^2+y^2=9$ to the hyperbola $\frac{x^2}{25}-\frac{y^2}{16}=1$, then tangents are at angle :

If the normals at $\left(x_i, y_i\right), i=1,2,3,4$ to the rectangular hyperbola $xy=2$ meet at the point $\left(3,4\right)$ , then $\frac{\sqrt{2}\sum t_1{t}_2{t}_3}{t_1{t}_2{t}_3{t}_4}$-

The portion of the line $lx+my=1$ intercepted by the circle $x^2+y^2=a^2$ subtends an angle of $45°$ at the centre of the circle. Prove that $4\left[a^2\left(l^2+m^2\right)-1\right]={\left[a^2\left(l^2+m^2\right)-2\right]}^2.$

If the tangent at the point $(\alpha ,\beta )$ to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ cuts the auxiliary circle at points with ordinates $\gamma$ and $\delta$, then prove that $\gamma ,\beta ,\delta$ are in Harmonic progression.

If $x+iy=\sqrt{(\varphi +i\psi )}$, where $\varphi$ and $\psi$ are real parameters, prove that $\varphi =$ constant and $\psi =$ constant represent two systems of rectangular hyperbolas which intersect at an angle of $\frac{\mathrm{\pi }}{6}$.

From any point of one hyperbola tangents are drawn to another which has th same asymptotes. Show that the chord of contact cuts off a constant area fro. the asymptotes.

Find the locus of the points of intersection of two tangents to a hyperbola $\frac{x^2}{25}-\frac{y^2}{16}=1.$ If sum of their slopes is constant $a$

A point $P$ moves such that the tangents $P{T}_1$ and $P{T}_2$ from it to the hyperbola $4x^2-9y^2=36$are mutually perpendicular. Then the equation of the locus of $P$is

Show that the line $4x-3y=9$ touches the hyperbola $4x^2-9y^2=27$.

Find the equation of tangent to the parabola $y^2=16x$ which is parallel to the line $3x-4y+5=0$. Also find the point of contact. 

The equation to the common tangents to the two hyperbolas $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$ are

$P$ is a point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1, N$ is the foot of the perpendicular from $P$ on the transverse axis. The tangent to the hyperbola at $P$ meets the transverse axis at $T$. If $O$ is the centre of the hyperbola, then $OT·ON$ is equal to

A common tangent to $9x^2-16y^2=144$ and $x^2+y^2=9$, is

The line $x\cos \alpha +y\sin \alpha =p$ touches the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, if

The value of $m$ for which $y=mx+6$ is a tangent to the hyperbola $\frac{x^2}{100}-\frac{y^2}{49}=1$ is