Basics of Ellipse

$P\left(x, y\right)$ is any point on the ellipse $x^2+4y^2=1$. Let $E, F$ be the foci of the ellipse.

Consider the following points :

$1.$ $\left(\frac{\sqrt{3}}{2}, 0\right)$

$2.$ $\left(\frac{\sqrt{3}}{2}, \frac{1}{4}\right)$

$3.$ $\left(\frac{\sqrt{3}}{2}, -\frac{1}{4}\right)$

Which of the above points lie on latus rectum of ellipse?

$P\left(x, y\right)$ is any point on the ellipse $x^2+4y^2=1$. Let $E, F$ be the foci of the ellipse.

What is $PE+PF$ equal to?

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

If $e$ is the eccentricity of the ellipse $4x^2+9y^2+8x+36y+4=0$, then $e=$

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

If the length of the latus-rectum of an ellipse is equal to the length of its semi-minor axis, then the eccentricity of the ellipse is

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

The curve represented by the parametric equation $x=2\left(\cos t+\sin t\right); y=5\left(\cos t-\sin t\right)$ is

If the distance between two foci is same as the length of a latus-rectum of an ellipse, then the ellipse has the eccentricity $e=$

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

If the length of the semi major axis is the same as the distance between two foci of an ellipse, then its eccentricity $e=$

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

Find the equation of ellipse with centre at origin, major axis on the $x$-axis and satisfying 
Length of major axis $=8$ and eccentricity$=\frac{1}{2}$

Find the equation of ellipse having foci at $(\pm 3,0),$ passing through$(4,1)$.

Find the equation of ellipse having foci at $(0,\pm 4), e=\frac{4}{5}$.

Find the equation of ellipse satisfying that the vertices at $(0,\pm 10),e=\frac{4}{5}$

Find the co-ordinates of the foci of the ellipse $\frac{x^2}{36}+\frac{y^2}{49}=1$