Tangent and Normal of 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

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

If a tangent of slope $2$ of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{1}=1$ passes through the point $\left(-\mathrm{2,0}\right)$, then the value of $a^2$ is equal to

Tangent at a point $P$ of ellipse $9x^2+16y^2-144=0$ having eccentric angle  $\theta =\frac{1}{2}{\sin }^{-1}\left(\frac{1}{7}\right)$ is drawn. Calculate $\angle PON$ if $N$ is the foot of perpendicular from centre $O$ of the ellipse to this tangent :

If the variable line $\mathrm{y}=\mathrm{k} \mathrm{x}+2 \mathrm{h}$ is tangent to an ellipse $2x^2+3y^2=6,$ then locus of $P(h, k)$ is a conic $C$. Find eccentricity of conic $C$.
 

The tangent to the ellipse$\frac{x^2}{32}+\frac{y^2}{18}=1$ having slope $m$ as $\frac{-3}{4}$ meets the co-ordinate axes at $A$ and $B$. Then the area of the $△AOB$, where $O$ is the origin, is equal to

The normal to the curve $2x^2+y^2=12$ at the point $(2,2)$meets the curve again at a point $P$.Then the point $P$ is

The equation of the normal to the ellipse  $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at the end of the latus rectum in first quadrant is:

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

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

Normal to the ellipse $\frac{x^2}{64}+\frac{y^2}{49}=1$ intersects the major and minor axis at $P$ and $Q$ respectively. Then the locus of the point dividing segment $PQ$ in $2:1$ internally, is

The ellipse $\frac{x^2}{9}+\frac{y^2}{b^2}=1$ has $\sqrt{3}x+y=4$ as normal to it, then value of $b$, if $(b\ne 1)$, is: