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If the straight-line x cos α + y sin α = p touches the curve x2/a2 + y2/b2 = 1, then prove that a2 cos2 α + b2 sin2 α = p2.

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Given: equation of straight line x cosα + y sinα = p, equation of curve x2/a2 + y2/b2 = 1 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

We know that, if a line y = mx+c touches the eclipse x2/a2 + y2/b2 = 1, 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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