### Ellipse

goals
If the polar with respect to $y^2=4x$ touches the ellipse $\displaystyle\frac{x^2}{{\alpha}^2}+\frac{y^2}{{\beta}^2}=1$, the locus of its pole is
Let $\displaystyle { F }_{ 1 }\left( { x }_{ 1 },0 \right)$ and $\displaystyle { F }_{ 2 }\left( { x }_{ 2 },0 \right)$, for $\displaystyle { x }_{ 1 }<0$ and $\displaystyle { x }_{ 2 }>0$, be the foci of the ellipse $\displaystyle \frac { { x }^{ 2 } }{ 9 } +\frac { { y }^{ 2 } }{ 8 } =1$. Suppose a parabola having vertex at the origin and focus at $\displaystyle { F }_{ 2 }$ intersects the ellipse at point $M$ in the first quadrant and at point $N$ in the fourth quadrant.
From point $A$ two tangents are drawn to ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$.If tangents meet coordinate axes at concyclic points then locus of point $A$ is
A coplanar beam of light emerging from a point source have equation $\lambda x-y+2(1+\lambda )=0,\lambda \in R$. The rays of the beam strike inner part of an elliptical surface and get reflected. The reflected rays form another convergent beam having equation $\mu x-y+2(1-\mu )=0,\mu \in R.$ Further it is found that the foot of the perpendicular from the point $(2, 2)$ upon any tangent to the ellipse lies on the circle $x^2+y^2-4y-5=0.$
An ellipse passing through the origin has its foci $(3, 4)$and $(6, 8)$. The length of its semi-mirror axis is b. Then the value of $\dfrac{b}{\sqrt 2}$ is