### Ellipse

goals
Let the equation of the ellipse be $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$. Let $f(x,y) = \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} - 1$. To determine whether the point $(x_1,y_1)$ lies inside the ellipse, the necessary condition is:
If the normal at the end of latus rectum of the ellipse $\frac{x^{2}}{a^{2}} + {y^{2}}{b^{2}}= 1$ passes through an extremity of minor axis, then $e^{4} +e^{2} -1=0$ or $e^{2} =\frac{\sqrt{5-1}}{2}$. A
$PG$ is the normal to a standard ellipse at $P$. $G$ being on the major axis. $GP$ is produced outwards to $Q$ so that $PQ=GP$. Show that the locus of $Q$ is an ellipse whose eccentricity is $\cfrac{{a}^{2}-{b}^{2}}{{a}^{2}+{b}^{2}}$

A ray of light along the line $x = 3$ is reflected at the ellipse $\dfrac{{{x^2}}}{{25}} + \dfrac{{{y^2}}}{{16}} = 1$. The slope of the reflected ray is

The angle between the lines $\dfrac{x}{a}+\dfrac{y}{b}=1$ and $\dfrac{x}{a}-\dfrac{y}{b}=1$ is.