Question

Find the condition for the line $x\cos \alpha +y\sin \alpha =p$ to be a tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Answer 1

Equation of tangent in parametric form to the given ellipse is written as

$\frac{x}{a}\cos \theta +\frac{y}{b}\sin \theta =1$

Comparing above equation with $x\cos \alpha +y\sin \alpha =p$

We get, $\frac{\cos \theta }{a\cos \alpha }=\frac{\sin \theta }{b\sin \alpha }=\frac{1}{p}$

$\Rightarrow \cos \theta =\frac{a\cos \alpha }{p}$ and $\sin \theta =\frac{b\sin \alpha }{p}$

Now square and add both,

$1=\frac{a^2{\cos }^2\alpha }{p^2}+\frac{b^2{\sin }^2\alpha }{p^2}$

$\Rightarrow a^2{\cos }^2\alpha +b^2{\sin }^2\alpha =p^2$ which is the required condition