Basics of Ellipse

The foci of the ellipse $16x^2+25y^2=400$ are -

Find the length of the focal chord of the ellipse $\frac{x^2}{4^2}+\frac{y^2}{3^2}=1$, which is inclined to the major axis at angle $60°$.

Find the length of the focal chord of the ellipse $\frac{x^2}{3^2}+\frac{y^2}{2^2}=1$, which is inclined to the major axis at angle $60°$.

Find the length of the focal chord of the ellipse $\frac{x^2}{3^2}+\frac{y^2}{2^2}=1$, which is inclined to the major axis at angle $30°$.

Find the length of the focal chord of the ellipse $\frac{x^2}{3^2}+\frac{y^2}{2^2}=1$, which is inclined to the major axis at angle $45°$.

The equation of the chord joining two points having eccentric angles $\theta \text{ and }\varphi$ an the ellipse is 

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

Find the equation of the ellipse which passes through the points $(-3, 1)$ and $(2, -2),$ whose center lies at $(0, 0)$ and major axis lies along the $x$-axis.

The point $(1, 3)$ with respect to the ellipse $4x^2+9y^2-16x-54y+61=0$ lies $\_\_\_\_\_\_\_$

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

Eccentricity of an ellipse is $\frac{2}{3}$, and length of its latus rectum is minimum value of the function, $f\left(\lambda \right)={\lambda }^2-\frac{4\lambda }{3}+\frac{40}{9}$, then the area (in sq. unit) of ellipse is

A variable point $P$ moves in $xy$ plane such that the sum of its distances from the points $\left(\sqrt{21},0\right)$ & $\left(-\sqrt{21},0\right)$ is $10$ and the line $y=2x+c$ always touch the path of the point $P$, then the value of $c$ is

The equation of an ellipse whose endpoints of the minor axis coincides with the foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ and the length of the major axis is equal to the diameter of the auxiliary circle of the ellipse $\frac{x^2}{16}+\frac{y^2}{3}=1$ is 

The parametric equation of the ellipse $x^2+4y^2-4x-8y+4=0$ is

The equation of the auxiliary circle of the ellipse $9x^2+4y^2-24x-36y+36=0$ is

If the coordinates of the centre of the circle passing through the foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ are $(0, 3)$, the radius of the circle is

The eccentric angle of a point in the first quadrant, which lies on the ellipse $x^2+3y^2=6$ and is $2$ unit away from the centre of the ellipse is

The sum of distances of any point on the ellipse $\frac{x^2}{4}+\frac{y^2}{3}=1$ from its two directrix is

Let $P$ be a variable point on the ellipse $\frac{x^2}{5+2\sqrt{3}}+\frac{y^2}{15}=1$ with foci $F_1$ & $F_2$. If the maximum area of the triangle $P{F}_1{F}_2$ is equal to one third the area of the ellipse. The eccentricity of the ellipse is