Ellipse practice · Mixed difficulty

The product of the perpendiculars drawn from the two foci of an ellipse to the tangent at any point of the ellipse, is

The area (in square units) of the quadrilateral formed by the tangents at the end-points of latusrecta to the ellipse $\frac{x^2}{9}+\frac{y^2}{5}=1$, is

If the length of the latusrectum of the ellipse ${x^2}{\tan ^2}\theta + {y^2}{\sec ^2}\theta = 1$ is $\frac{1}{2}$, then $\theta =$

If $x$ and $y$ are real numbers such that $\frac{(x-4)^2}{4}+\frac{y^2}{9}=1$, then the largest possible value of $\frac{x^2}{4}+\frac{y^2}{9}$ is

The sum of the squares of the perpendicular on any tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ from two points on the minor axis, each at a distance $\sqrt{a^2-b^2}$ from the centre, is

The area of the triangle formed by three points on the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ whose eccentric angles are $\alpha ,\beta$ and $\gamma$ is

The equation $\frac{x^2}{2-r}+\frac{y^2}{r-5}+1=0$ represent the ellipse if

In an ellipse the distance between the foci is 8 and the distance between the directrices is 25. The length of major axis is

The sum of the focal distances of any point on the conic $\frac{{{x^2}}}{{25}} + \frac{{{y^2}}}{{16}} = 1$ is

If $Q$ is the point on the auxiliary circle corresponding to a point $P$ on an ellipse.
Then, the normals at $P$ and $Q$ meet on

A tangent at any point to the ellipse $4{x^2} + 9{y^2} = 36$ is cut by the tangent at the extremities of the major axis at $T$ and $T'$. The circle on $TT'$ as diameter passes through the point

The eccentricity of the ellipse $4 x^2+9 y^2=36$, is

The value of $k$, if $\left( {1,2} \right),$ $\left( {k, - 1} \right)$ are conjugate points with respect to the ellipse $2{x^2} + 3{y^2} = 6,$ is

The foci of the ellipse, $25{\left( {x + 1} \right)^2} + 9{\left( {y + 2} \right)^2} = 225$ are at,

Let the tangent and normal at the point $(3 \sqrt{3}, 1)$ on the ellipse $\frac{x^{2}}{36}+\frac{y^{2}}{4}=1$ meet the $y$-axis at the points $A$ and $B$ respectively. Let the circle $C$ be drawn taking $A B$ as a diameter and the line $x=2 \sqrt{5}$ intersect $C$ at the points $P$ and $Q$. If the tangents at the points $P$ and $Q$ on the circle intersect at the point $(\alpha, \beta)$, then $\alpha^{2}-\beta^{2}$ is equal to

If ${e_1}$ and ${e_2}$ are the eccentricities of the ellipse, $\frac{{{x^2}}}{{18}} + \frac{{{y^2}}}{4} = 1$ and the hyperbola, $\frac{{{x^2}}}{9} - \frac{{{y^2}}}{4} = 1$ respectively and $\left( {{e_1},{e_2}} \right)$ is a point on the ellipse, $15{x^2} + 3{y^2} = k$, then $k$ is equal to

If ${e_1}$ and ${e_2}$ are respectively the eccentricities of the ellipse $\frac{{{x^2}}}{{18}} + \frac{{{y^2}}}{4} = 1$ and the hyperbola $\frac{{{x^2}}}{9} - \frac{{{y^2}}}{4} = 1,$ then the relation between ${e_1}$ and ${e_2}$ is

If the polar with respect to ${y^2} = 4ax$ touches the ellipse $\frac{{{x^2}}}{{{\alpha ^2}}} + \frac{{{y^2}}}{{{\beta ^2}}} = 1,$ the locus of its pole is

If the normals at the points $A, B, C$ and $D$ of an ellipse meet in a point $O$, then $S A \cdot S B \cdot S C \cdot S D$ is equal to (where, $S$ is one of the foci)

The eccentricity of the conic $4{x^2} + 16{y^2} - 24x - 32y = 1$ is

The equation of normal at the point $\left( {0,3} \right)$ of the ellipse $9{x^2} + 5{y^2} = 45,$ is

The locus of the mid-points of chords of the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ that touch the circle ${x^2} + {y^2} = {b^2},$ is

If $lx + my + n = 0$ represents a chord of the ellipse ${b^2}{x^2} + {a^2}{y^2} = {a^2}{b^2}$ whose eccentric angles differ by ${90^0}$, then

The locus of the middle points of chords of an ellipse which pass through a fixed point is

In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at $(0,5\sqrt 3 ),$ then the length of its latus rectum is

The tangent at a point $P(\theta)$ to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ cuts the auxiliary circle at points $Q$ and $R$. If $Q R$ subtends a right angle at the centre $C$ of the ellipse, then the eccentricity of the ellipse, is

The eccentricity of the conic $4 x^2+16 y^2-24 x-32 y=1$ is

If equation of the ellipse is $2 x^2+3 y^2-8 x+6 y+5=0$, then which of the following is true?

An ellipse whose focii are $(2,4)$ and $(14,9)$ and touches x-axis, then its eccentricity is

If $C$ is the centre of the ellipse $9{x^2} + 16{y^2} = 144$ and $S$ is one focus. The ratio of $CS$ to major axis, is