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

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 \( SQ \times S’Q’ = b^2\).

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