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.