Pair of Tangents, Chord of Contact and Chord with Midpoint of a Hyperbola

The locus of a variable point whose chord of contact w.r.t. the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ subtends a right angle at the origin is

If any tangent to the parabola $x^2=4y$ intersects the hyperbola $xy=2$ at two points $P$ and $Q,$ then the mid-point of line segment $PQ$ lies on a parabola with axis along

Equation of the chord of the hyperbola $25{\mathrm{x}}^2-\mathrm{ }16{\mathrm{y}}^2=400$ which is bisected at the point $\left(6,\mathrm{ }2\right)$ , is -

Find the eccentricity of conjugate hyperbola of the hyperbola $16x^2+12y–44=3y^2+32x$.

Find the locus of middle points of parallel chords with gradiant $-1$ of rectangular hyperbola $xy=8$.

If the line $x=9$, is the chord of contact of the hyperbola $x^2-y^2=9$, then the equation of the corresponding pair of tangents is _______.

If $125x-48y=k+480$ is the equation of the chord of the hyperbola $25x^2-16y^2=400$ which is bisected at $\left(5, 3\right)$, then the value of $k$ is:

Cotyledons are also called-

If the chords of the hyperbola $x^2-y^2=a^2$ touch the parabola $y^2=4ax$ . Then, the locus of the middle points of these chords is

If x = 9 is a chord of contact of the hyperbola $x^2-y^2=9$ , then the equation of the tangent at one of the points of contact is

From any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,$ tangents are drawn to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=2.$ The area cut off by the chord of contact on the region between the asymptotes is equal to -

The equation of the chord of the hyperbola $25x^2-16y^2=400$ which is bisected at $\left(5,3\right)$ is $125x-48y=k+480,$ then the value of $k$ is

The angle between lines joining origin to the points of intersection of the line $x\sqrt{3}+y=2$ and the curve $y^2-x^2=4$ is

The locus of a point whose chord of contact w.r.t. the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ touches the circle described on the straight line joining the foci of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$as diameter is

What will be equation of that chord of hyperbola $25x^2-16y^2=400$ whose midpoint is $\left(5,3\right)$?

What will be equation of that chord of hyperbola $25x^2-16y^2=400,$ whose midpoint is $\left(5,3\right)$

The locus of a point, from where tangents to the rectangular hyperbola $x^2-y^2=a^2$ contain an angle of ${45}^o$ is

The locus of the middle points of chords of hyperbola $3x^2-2y^2+4x-6y=0$ parallel to $y=2x$ is

Locus of the middle points of the chords of the hyperbola $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}-\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=\mathrm{ }1$ which passes through a fixed point $\left(\mathrm{\alpha },\mathrm{ }\mathrm{\beta }\right)$ is hyperbola whose centre is -

Equation of the chord of the hyperbola $25{\mathrm{x}}^2-\mathrm{ }16{\mathrm{y}}^2=400$ which is bisected at the point $\left(6,\mathrm{ }2\right)$ , is -