Parametric Equations of an Ellipse

Find the eccentric angle of a point on the ellipse $\frac{x^2}{6}+\frac{{\displaystyle y^2}}{2}=1$, whose distance from centre of the ellipse is $\sqrt{5}$.

The distance of a point $P$ on the ellipse $\frac{x^2}{6}+\frac{y^2}{2}=1$ from centre $C$ is $2$ , then eccentric angle of $P$ may be

${\alpha }_1, {\alpha }_2, {\alpha }_3, a_4$ are the eccentric angles of four concyclic points on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. Then prove that ${\alpha }_1+{\alpha }_2+{\alpha }_3+{\alpha }_4=2n\pi , n\in \mathrm{\mathbb{N}}$.

The eccentric angle of an end point of a focal chord of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is $\varphi$. Show that the eccentric angle of other end point of the chord is $2 {\tan }^{-1}\left(\frac{e-1}{e+1}\cot \frac{\varphi }{2}\right)$ or, $2 {\tan }^{-1}\left(\frac{e+1}{e-1}\cot \frac{\varphi }{2}\right)$.

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

The eccentric angles of two points $L$ and $M$ on $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, \left(a^2>b^2\right)$ are $(\alpha +\beta )$ and $(\alpha -\beta )$ respectively; prove that the equation of the chord $LM$ is $\frac{x}{a}\cos \alpha +\frac{y}{b}\sin \alpha =\cos \beta$.

Coordinates of a point on the ellipse $9x^2+16y^2=144$ are $\left(2, \frac{3\sqrt{3}}{2}\right)$. Find the eccentric angle of the point.

Find the eccentric angle of the point $P\left(\frac{1}{\sqrt{2}}, \frac{\sqrt{3}}{2}\right)$ on the ellipse $x^2+2y^2=2$.

Find the eccentric angles of the end points of a latus-rectum of the ellipse $2x^2+4y^2=1$.

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

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 coordinates of points $A$ and $B$ shown on a ellipse, whose equation is given by $4x^2+9y^2=36,$ are :-

If the reflection of the ellipse $\frac{(x-4{)}^2}{16}+\frac{(y-3{)}^2}{9}=1$ in the line mirror $x-y-2=0$ is $k_1{x}^2+k_2{y}^2-160x-36y+292=0,$ then $k_1+k_2$ is equal to

Suppose $x$ and $y$ are real numbers and that $x^2+9y^2-4x+6y+4=0$, then the minimum value of $\left(4x-9y\right)$ is 

The area of the triangle inscribed in an ellipse bears the ratio as $\sqrt{5}:3$ to the area of the triangle formed by joining the points on the auxiliary circle corresponding to the vertices of the first triangle, then the eccentricity of the ellipse is :

The line passing through the extremities $A$ and $B$ of the major and minor axis respectively of an ellipse $x^2+9y^2=9$ meets it's auxiliary circle at the point $M$. Then the area of the triangle formed by the vertices $A, M$ and the origin is 

Find $\Sigma \cos \alpha \cos \beta +\Sigma \sin \alpha \sin \beta$, If circumcentre of an equilateral triangle inscribed in $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$ with vertices having eccentric angles $\alpha ,\beta ,\gamma$ respectively is $\left(x_1,y_1\right)$.

For the given ellipse $\frac{x^2}{9}+\frac{y^2}{16}=1$, equation of auxiliary circle is _____

The eccentric angle of a point on the ellipse $\frac{x^2}{4}+\frac{y^2}{3}=1$ at a distance of $\frac{5}{4}$ units from the focus on the positive $x$-axis :

The locus of the point which divides the double ordinates of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ in the ratio $1:2$ internally is :