Ellipse practice · Mixed difficulty

An ellipse slides between two perpendicular lines. The locus of its centre, is

The foci of the conic $25 x^2+16 y^2-150 x=175$ are:

An ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$ passes through the point $(-3,1)$ and its eccentricity is $\sqrt{\frac{2}{5}}$. Then length of latus rectum is

The locus of the point of intersection of perpendicular tangents to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is

The length of the latus rectum of the ellipse $9 x^2+4 y^2=1$ is

Number of tangents from $\left( {7,6} \right)$ to ellipse $\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{{25}} = 1$ is

Tangents are drawn to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$ and the circle $x^2+y^2=a^2$ at the points where a common ordinate cuts them (on the same side of the $x$-axis). Then, the greatest acute angle between these tangents is given by

The foci of an ellipse are $\left( {0, \pm 6} \right)$ and the equations of the directrices are $y = \pm 9.$ The equation of the ellipse is

The normal at a variable point $P$ on an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ of eccentricity e meets the axes of the ellipse in $Q$ and $R$, then the locus of the mid-point of $Q R$ is a conic with eccentricity $e^{\prime}$ such that

The equation of the circle passing through the foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ and having centre at $(0,3)$, is

Tangent to the ellipse $\frac{{{x^2}}}{{32}} + \frac{{{y^2}}}{{18}} = 1$ having slope $- \frac{3}{4}$ meet the coordinate axes in $A$ and $B$. Find the area of the $\Delta AOB,$ where $O$ is the origin

Which one of the following is the common tangent to the ellipses, $\frac{x^2}{a^2+b^2}+\frac{y^2}{b^2}=1 \& \frac{x^2}{a^2}+\frac{y^2}{a^2+b^2}=1$ ?

The locus of the point of intersection of tangents to an ellipse at two points whose eccentric angle differ by a constant $\alpha$ is $\frac{x^2}{\lambda \sec ^2\left(\frac{\alpha}{2}\right)}+\frac{y^2}{\mu \sec ^2\left(\frac{\alpha}{2}\right)}=1$ then $\lambda+\mu$

The tangent drawn from $\left( {\mathrm{\alpha },\mathrm{\beta }} \right)$ 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 $\left( {\mathrm{\alpha },\mathrm{\beta }} \right)$ is

The point on the ellipse $4 x^2+9 y^2=1$ where the tangent drawn is parallel to line $8 x=9 y$ is

$P$ and $Q$ are the foci of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $B$ is an end of the minor axis. If $P B Q$, is an equilateral triangle, then the eccentricity of the ellipse is

The tangent to the ellipse $3 x^2+16 y^2=12$, at the point $\left(1, \frac{3}{4}\right)$ intersects the curves $y^2+x=0$ at:

A circle has the same center as an ellipse and passes through the foci $F_1$ and $F_2$ of the ellipse, such that two curves intersect at four points. Let $P$ be any one of their point of intersection. If the major axis of the ellipse is 17 and the area of triangle $P F_1 F_2$ is 30 , then the distance between the foci is

If $(a \cos \alpha, b \sin \alpha),(a \cos \beta, b \sin \beta)$ are the end points of a focal chord, then which of the following is correct?

Equation of the line passing through the points of intersection of the parabola $x^{2}=8 y$ and the ellipse $\frac{x^{2}}{3}+y^{2}=1$ is :

In an ellipse, the distances between its foci is 6 and minor axis is 8. Then, its eccentricity is

If the distance between the foci of an ellipse is equal to its minor axis, then its eccentricity is

Area bounded between $4 x^{2}+y^{2}-8 x+4 y-4=0$ is

If $A,A'$ are the vertices, $S,S'$ are the foci and $Z,Z'$ are the feet of the directrices of an ellipse with centre $C$, then $CS,CA,CZ$ are in

If $P(x, y), F_1(3,0), F_2(-3,0)$ and $16 x^2+25 y^2=400$, then $P F_1+P F_2$ equals

Locus of all such points from where the tangents drawn to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ are always inclined at $45^{\circ}$, is

A point on the ellipse, $4 x^{2}+9 y^{2}=36$, where the normal is parallel to the line, $4 x-2 y-5=0$, is :

The locus of the mid-points of the chords of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ whose poles are on the auxiliary circle on the tangents at the extremities of which intersect on the auxiliary circle, is

The equation of the ellipse whose axes are coincident with the coordinate axes and which touches the straight line $3x-2y-20 = 0$ and $x+6 y-20=0$, is

If $\left( {\sqrt 3 } \right)bx + ay = 2ab$ touches the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$, then eccentric angle $\phi$ is