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.

Find the equation of the diameter of an ellipse $2x^2+3y^2=6$ conjugate to the diameter $3y+2x=0$

Find the equation of the diameter of an ellipse $3x^2+4y^2=5$ conjugate to the diameter $y+3x=0$

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.

From any point on the line $\left(t+2\right)\left(x+y\right)=1, t\ne -2,$ tangents are drawn to the ellipse $4x^2+16y^2=1.$ It is given that chord of contact passes through a fixed point. Then the number of integral values of $\text{'}t\text{'}$ for which the fixed point always lies inside the ellipse is

Which of the following options is most revalent?

Statement 1:
Let $P$ be any point on a directrix of an ellipse. Then, the chords of contact of the point $P$ with respect to the ellipse and its auxiliary circle intersect at the corresponding focus.

Statement 2: 
The equation of the family of lines passing through the point of intersection of lines $L_1=0$ and $L_2=0$ is $L_1+\lambda L_2=0$.

 If tangent to parabola $y^2=4x$ intersect the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$ at $A$ and $B$ and locus of point of intersection of tangents at $A$ and $B$ is a conic $C$, then

If line $x+y=1$ is intersecting ellipse $\frac{x^2}{3}+\frac{y^2}{4}=1$ at two distinct points $A$ and $B$, then point of intersection of tangents at $A$ and $B$ :

Cotyledons are also called-

From the point $P\left(\mathrm{3,4}\right)$ pair of tangents $PA$ and $PB$ are drawn to the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1.$ If $AB$ intersects $y$-axis at $C$ and $x$-axis at $D$, then $OC·OD$ is equal to (where $O$ is the origin)

Let from a point $A\left(h,k\right)$ chords of contact are drawn to the ellipse $x^2+2y^2=6$ where all these chords touch the ellipse $x^2+4y^2=4$. Then, the perimeter (in units) of the locus of point $A$ is

The line $2\mathrm{x}+\mathrm{y}=3$ interesects the ellipse $4{\mathrm{x}}^2+{\mathrm{y}}^2=5$ at two points. The point of intersection of the tangents to the ellipse at these points 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