Basics of Ellipse

A rod of given length moves such that its extremities lie on two fixed perpendicular lines. Any point (other than mid point) on the rod describes

If the focal distance of an end of the minor axis of an ellipse, whose axes along the $x$ and $y$ axes respectively is $k$ and the distance between the foci is $2 h$. Then the equation of the ellipse is

If $e_1$ be the eccentricity of the conic $9x^2+4y^2=36$ and $e_2$ is the eccentricity of the conic $9x^2-4y^2=36$ and $e_3$ be the eccentricity of conic $x^2-y^2=36$ then $18\left(e_1^2+e_2^2\right)·e_3^2$   is equal to -

The eccentricity of the ellipse, which passes through the points $(2,-2)$ and $(-3,1)$ is

If the eccentricity of an ellipse is $\frac{1}{6}th$ of the distance between the foci and distance between the foci is equal to $4\sqrt{2}$ units, then find the length of the latus rectum of the ellipse.

Find the equation of the ellipse if its eccentricity is $\frac{\sqrt{13}}{7}$ and co-ordinates of the foci are $(0, \pm 2\sqrt{13}).$

Find the length of major and minor axis, co-ordinates of the vertices and the foci, eccentricity and length of the latus rectum of the ellipse $y^2+36x^2=36$

Find the length of the latus rectum and the equation of an ellipse, the ends of whose major axis are $\left(\pm 9, 0\right)$ and the ends of whose minor axis are $(0, \pm 6).$

If the eccentricity of an ellipse is $\frac{\sqrt{13}}{7}$ and distance between its foci is $2\sqrt{13},$ then find the length of the latus rectum of the ellipse.

The length of the semi-minor axis of an ellipse with $x$-axis as the major axis is $3$ units and the distance of the foci from the centre is $2\sqrt{5}.$ Find the equation of the ellipse.

If $\frac{x^2}{64}+\frac{y^2}{36}=1$ represents the equation of an ellipse, then find the distance between its foci.

Find the $co$-ordinates of the foci and the vertices of the ellipse $x^2+4y^2=16.$

Find the equation of the ellipse with centre at the origin, major-axis on the $y-$axis and passing through the points $(2, 0)$ and $(0, 3).$

An iron rod of given length moves with its extremities on two fixed straight lines at right angles. If $P(x, y)$ is any point on the iron rod, then which section of the cone is represented by the locus of the point $P.$

If $\mathrm{\alpha }$ and $\mathrm{\beta }$ are the eccentric angles of the extremities of a focal chord of an ellipse, then the eccentricity of the ellipse is

If the line $px+qy=r$ intersects the ellipse $x^2+4y^2=4$ in points, whose eccentric angles differ by $\frac{\mathrm{\pi }}{3},$ then $r^2$ is equal to

Length of the latus-rectum of the ellipse represented by $x=3(\cos t+\sin t), y=4(\cos t-\sin t);$ is given by

The co-ordinates of a focus of the ellipse $4x^2+9y^2=1,$ is

If the distance between foci is $2$ and the distance between the directrices is $5,$ then equation of the ellipse in the standard form is

Let $P$  be a variable point on the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ with foci at $F_{1 }$ & $F_2$