Equation of an Ellipse

The parametric equations of the ellipse whose foci are $\left(-3,0\right),\left(9,0\right)$ and eccentricity is $\frac{1}{3}$, are

If the eccentricity and the length of the latus rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ are $\frac{\sqrt{3}}{2}$ and $1$ respectively, then the sum of the lengths of major axis and minor axis of the ellipse is

$P\left({\theta }_1\right)$ and $Q\left({\theta }_2\right)$ are two points on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with eccentricity $e$. If $PSQ$ is a focal chord and $\tan \left(\frac{{\theta }_1}{2}\right)\tan \left(\frac{{\theta }_2}{2}\right)=-\left(2\sqrt{2}+3\right)$, then $e$ and $S$ are

For the ellipse $4{\left(x-2y+1\right)}^2+9{\left(2x+y+2\right)}^2=25,$ which of the following is true?

An ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b$ and the parabola $x^2=4\left(y+b\right)$ are such that the two foci of the ellipse and the end points of the latusrectum of parabola are the vertices of a square. The eccentricity of the ellipse is

An ellipse with its minor and major axis parallel to the coordinate axes passes through $\left(0,0\right),\left(1,0\right)$ and $\left(0,2\right)$. One of its foci lies on the $Y$-axis. The eccentricity of the ellipse is

If an ellipse has its foci at $\left(2,0\right)$ and $\left(-2,0\right)$ and its length of the latus rectum is $6$, then the equation of the ellipse is

Let $P$ and $Q$ be two points on the curves $x^2+y^2=2$ and $\frac{x^2}{8}+\frac{y^2}{4}=1$ respectively. Then the minimum value of the length $PQ$ is

If $\tan {\theta }_1·\tan {\theta }_2=-\frac{a^2}{b^2},$ then the chord joining two points $P\left(a\cos {\theta }_1,b\sin {\theta }_1\right)$ and $Q\left(a\cos {\theta }_2,b\sin {\theta }_2\right)$on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ will subtend a right angle at

A man running round a race course notes that the sum of the distances of two flag-posts from him is always $10 m$ and the distance between the flag-posts is $8 m.$ The area of the encloses field (in square meters) by his path is

The centre of the ellipse $\frac{{\left(x+y-3\right)}^2}{9}+\frac{{\left(x-y+1\right)}^2}{16}=1$ is

An ellipse, with foci at $\left(\mathrm{0,2}\right)$ and $\left(0,-2\right)$ and minor axis of length $4$ , passes through which of the following points?

The sum of distances of any point on the ellipse $\frac{x^2}{4}+\frac{y^2}{3}=1$ from its two directrix 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 $\left(\mathrm{0,5}\sqrt{3}\right),$ then the length of its latus rectum is:

If the length of the major axis of an ellipse is $3$ times its minor axis, then eccentricity of the ellipse is

Consider an ellipse, whose centre is at the origin and its major axis is along the $x-axis$ . If its eccentricity is $\frac{3}{5}$ and the distance between its foci is$6$ , then the area (in sq. units) of the quadrilateral inscribed in the ellipse, with the vertices as the vertices of the ellipse, is:

The eccentricity of the ellipse $9x^2+25y^2=225$ is

If the foci and vertices of an ellipse be $\left(\pm 1,0\right)$ and $\left(\pm 2, 0\right)$ then the minor axis of the ellipse is

The length of the latus - rectum of the ellipse $5x^2+9y^2=45$ is

The equation of an ellipse whose eccentricity is $\frac{1}{2}$ and the vertices are, $\left(4,0\right)$ and $\left(10, 0\right)$ is