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