# Ellipse Discriminant Eccentricity

Discriminant can be used to ellipses for identifying the status of their intersections in conjunction with eccentricity.
The eccentricity of $$\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ is $$\displaystyle e^2 = 1 – \frac{b^2}{a^2}$$.

### Worked Example of Ellipse Discriminant Eccentricity

(a)    The tangent $$\ell$$ has equation $$y=mx+k$$. Show that $$a^2m^2 + b^2 = k^2$$.

(b)    Show that the shortest distance from $$S$$ to $$\ell$$ is $$\displaystyle SQ=\frac{|mae + k|}{\sqrt{1+m^2}}$$.

(c)    Prove that $$SQ \times S’Q’ = b^2$$.