Math Help - Find the condition for line to touch a curve?

1. Find the condition for line to touch a curve?

Find the condition that the line $x\cos \alpha + y\sin \alpha = p$ may touch the curve

$\left(\frac{x - a}{a}\right)^{\frac{n}{n - 1}} + \left(\frac{y - b}{b}\right)^{\frac{n}{n - 1}} = 1$

2. Originally Posted by fardeen_gen
Find the condition that the line $x\cos \alpha + y\sin \alpha = p$ may touch the curve

$\left(\frac{x - a}{a}\right)^{\frac{n}{n - 1}} + \left(\frac{y - b}{b}\right)^{\frac{n}{n - 1}} = 1$
Assume $n>1,\ a, b>0$

The second curve is a hyper-ellipse, and the condition of tangency is that the equation you get by substituting $y$ from the equation for the line into the equation of the hyper-ellipse has a single real root.

CB