Answer :

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


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