Tangent and Normal to an Ellipse

The equation of the normal at the point $(2,3)$ to the ellipse $9x^2+16y^2=180$ is 

Equations of the common tangent of the ellipse $\frac{(x+1{)}^2}{2^2}+\frac{(y-1{)}^2}{3^2}=1$ and the circle $(x+1{)}^2+(y-1{)}^2=4$ are

The product of the perpendiculars drawn from the foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{25}=1$ upon the tangent to it at the point $\left(\frac{3}{2},\frac{5\sqrt{3}}{2}\right)$, is

Equation of the ellipse with axes coinciding the co-ordinate axes touches the straight lines $3\mathrm{x}\mathrm{ }-\mathrm{ }2\mathrm{y}\mathrm{ }-\mathrm{ }20=0$ and $\mathrm{x}\mathrm{ }+\mathrm{ }6\mathrm{y}\mathrm{ }-\mathrm{ }20=0$ , is:

The number of values of '$c$' such that the straight line $y=4x+c$ touches the curve $\frac{x^2}{4}+y^2=1$, is

If the normal at an end of a latus rectum of an ellipse passes through one extremity of the minor axis, then the eccentricity of the ellipse is given by

The equation of the tangents to the ellipse $9{\mathrm{x}}^2+\mathrm{ }16{\mathrm{y}}^2=144$ from the point $\left(2,\mathrm{ }3\right)$ are -

Equation of the ellipse with axes coinciding the co-ordinate axes touches the straight lines $3\mathrm{x}\mathrm{ }-\mathrm{ }2\mathrm{y}\mathrm{ }-\mathrm{ }20=0$ and $\mathrm{x}\mathrm{ }+\mathrm{ }6\mathrm{y}\mathrm{ }-\mathrm{ }20=0$ , is:

Equation of tangents to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$, which are perpendicular to the line $3x+4y=7$, are

The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latus ractum to the ellipse $\frac{x^2}{9}+\frac{y^2}{5}=1,$ is

Consider the curve $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. The portion of the tangent at any point of the curve intercepted between the point of contact and the directrix subtends at the corresponding focus an angle of

The length of the minor axis (along $y$-axis) of an ellipse in the standard form is $\frac{4}{\sqrt{3}}$. If this ellipse touches the line, $x+6y=8$; then its eccentricity is:

Let the line $y=mx$ and the ellipse $2x^2+y^2=1$ intersect at a point $P$ in the first quadrant. If the normal to this ellipse at $P$ meets the co-ordinate axes at $\left(-\frac{1}{3\sqrt{2}},0\right)$ and $\left(0,\beta \right)$, then $\beta$ is equal to

The equation of the tangent to the ellipse $5x^2+9y^2=45$, and perpendicular to the line $3x+2y+1=0$ is

Consider a tangent to the ellipse $\frac{x^2}{2}+\frac{y^2}{1}=1$ at any point. The locus of the mid-point of the portion intercepted between the axes is

The product of the perpendicular distances drawn from the points $(3,0)$ and $(-3,0)$ to the tangent of an ellipse $\frac{x^2}{36}+\frac{y^2}{27}=1$ at $\left(3,\frac{9}{2}\right)$ is

For the ellipse $\frac{x^2}{18}+\frac{y^2}{32}=1,$ if a tangent with slope $\frac{-4}{3}$ intersects the major and minor axes at $P$ and $Q$ respectively, find $P$ and $Q$

Find the condition for the line $ax+by+c=0$ to be a normal to an ellipse $\frac{x^2}{4}+\frac{y^2}{36}=1$

Slope of normal to the ellipse at a point$\mathrm{P}$ is $\frac{3}{4}$ and eccentricity of ellipse is $\frac{1}{3}$. If this normal makes acute angle$\prime \mathrm{\beta }\prime$ with its focal chord through $\mathrm{P}$, then $\mathrm{sin\beta }$ is

Let the ellipse, $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b,$ pass through the point $(2,3)$ and have eccentricity equal to $\frac{1}{2}.$ Then equation of the normal to this ellipse at $(2,3)$ is