Find the condition for line to touch a curve?

**fardeen_gen** · May 11th 2009, 10:38 PM

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

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

**CaptainBlack** · May 16th 2009, 01:03 AM

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

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