Given: equation of straight line x cosα + y sinα = p, equation of curve and the straight line touches the curve

To prove: a² cos²α + b² sin²α = p²

Explanation: given equation of the line is

x cosα + y sinα = p

⇒ y sinα = p-x cosα

Comparing this with the equation y = mx+c we see that the slope and intercept of the given line is

and

We know that, if a line y = mx+c touches the eclipse, then required condition is

c² = a²m²+b²

Now substituting the corresponding values, we get

Cancelling the like terms we get

p² = b²sin²α+a²cos²α

Hence proved