Basics of Ellipse

Find the eccentricity, foci and directrices of the ellipse:

$4x^2+y^2-8x+2y+1=0$.

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 $45°$.

Suppose that the chord joining the points ${\theta }_1\text{ and }{\theta }_2$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ intersects the major-axis at $(k,0)$. Show that $\tan \left(\frac{{\theta }_1}{2}\right)\tan \left(\frac{{\theta }_2}{2}\right)=\frac{k-a}{k+a}$.

Suppose that the chord joining the points ${\theta }_1\text{ and }{\theta }_2$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ intersects the major-axis at $(h,0)$. Show that $\tan \left(\frac{{\theta }_1}{2}\right)\tan \left(\frac{{\theta }_2}{2}\right)=\frac{h-a}{h+a}$.

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

If $\alpha \text{ and }\beta$ are the eccentric angles of the extremities of a focal chord of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$. Then the eccentricity of the ellipse.

If ${\theta }_1,{\theta }_2$ are the eccentric angles of the extremities of a focal chord of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$ and its eccentricity is $e$. Then show that $e\cos \left(\frac{{\theta }_1+{\theta }_2}{2}\right)=\cos \left(\frac{{\theta }_1-{\theta }_2}{2}\right)$.

Find the equation of the ellipse whose axes are parallel to the coordinate axes and centre is $\left(1,1\right)$ and passes through two given points $(4,3)\text{ and }(-1,4)$.

Draw the following standard equation of an ellipse $\frac{x^2}{49}+\frac{y^2}{16}=1$ using coordinates of vertices, coordinates of foci and equation of directrices. Then find the eccentricity of the ellipse.

Draw the following standard equation of an ellipse $\frac{x^2}{144}+\frac{y^2}{81}=1$ using coordinates of vertices, coordinates of foci and equation of directrices. Then find the eccentricity of the ellipse.

Draw the following standard equation of an ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ using coordinates of vertices, coordinates of foci and equation of directrices. Then find the eccentricity of the ellipse.

Draw the following standard equation of an ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ using coordinates of vertices and foci. Then find the eccentricity of the ellipse.

Draw the following standard equation of an ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ using coordinates of vertices and foci. Then find the eccentricity of the ellipse.

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 $(0,\pm 10),e=\frac{4}{5}$