Pair of Tangents, Chord of Contact and Chord with Midpoint of an Ellipse

Tangents are drawn to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at points where it is intersected by the line $lx+my+n=0$. Find the point of intersection of tangents at these points.

If the lines $2x-y+3=0$ and $4x+ky+3=0$ are conjugate with respect to the ellipse $5x^2+6y^2-15=0$, then $\text{'}k\text{'}$ equals

The line $2x+y=3$ intersects the ellipse $4x^2+y^2=5$ at two points. The tangents to the ellipse at these two points intersect at the point.

Let from a point $A\left(h,k\right)$ chord of contacts of the tangents are drawn to the ellipse $x^2+2y^2=6$ such that all these chords touch the ellipse $x^2+4y^2=4$, then the locus of the point $A$ is

$AB$ is a diameter of $x^2+ 9y^2=25$ . The eccentric angle of $A$ is $\pi /6$. Then the eccentric angle of $B$ is -

If the point of intersection of the ellipses $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$ and $\frac{{\mathrm{x}}^2}{{\mathrm{\alpha }}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{\beta }}^2}=1$ be at the extremities of the conjugate diameters of the former, then -

Tangents are drawn from the points on the line $\mathrm{x}\mathrm{ }-\mathrm{ }\mathrm{y}\mathrm{ }-\mathrm{ }5=0$ to ${\mathrm{x}}^2+\mathrm{ }4{\mathrm{y}}^2=4$ , then all the chords of contact pass through a fixed point, whose co-ordinates are -

The length of the diameter of the ellipse $\frac{{\mathrm{x}}^2}{25}+\frac{{\mathrm{y}}^2}{9}=1$ perpendicular to the asymptotes of the hyperbola $\frac{{\mathrm{x}}^2}{16}-\frac{{\mathrm{y}}^2}{9}=1$ passing through the first and third quadrant is

The chord of contact of the tangents drawn from $(\alpha ,\beta )$ to an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ touches the circle $x^2+y^2=c^2$, then the locus of $(\alpha ,\beta )$ is

The chord of contact of the tangents drawn from $(\mathrm{\alpha }, \mathrm{\beta })$ to an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ touches the circle $x^2+y^2=c^2$, then the locus of $(\mathrm{\alpha }, \mathrm{\beta })$ is:

If the chords of constant of tangents from two points $(x_1,y_1)$ and $(x_2,y_2)$ to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ are at right angles, then $\frac{x_1{x}_2}{y_1{y}_2}$ is equal to

If a variable tangent of the circle $x^2+y^2=1$ intersects the ellipse $x^2+{2y}^2=4$ at points $P$ and $Q$, then the locus of the point of intersection of tangent at $P$ and $Q$ is

If $2y=x$ and $3y+4x=0$ are the equations of a pair of conjugate diameters of an ellipse, then the eccentricity of the ellipse is

If the chords of contact of tangents from two points $\left(x_1,y_1\right)$ & $\left(x_2,y_2\right)$ to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ are at right angles then $\left(\frac{x_1{x}_2}{y_1{y}_2}\right)$ is equal to