If the straight line `xcosalpha+ysinalpha=p` touches the curve `(x^2)/(a^2)+(y^2)/(b^2)=1`, then prove that `a^2\ cos^2alpha+b^2\ sin^2alpha=p^2`.

Explanation

To prove that \( a^2 \cos^2 \alpha + b^2 \sin^2 \alpha = p^2 \) given that the line \( x \cos \alpha + y \sin \alpha = p \) touches the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), we can follow these steps:

### Step 1: Set Up the Problem
We start with the line equation:
\[ x \cos \alpha + y \sin \alpha = p \]
This line touches the ellipse given by:
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