Find the condition for the line x cos α + y sin α = p to be a tangent to the ellipse x2/a2 +y2/b2 = 1

Solution:

Given equation of line is x cos α + y sin α = p

=> y sin α = -x cos α + p

=> y = -x (cos α/sin α) + p/sin α

This is of the form y = mx+c.

Comparing, we get m = -(cos α/sin α) and c = p/sin α.

If y = mx+c is a tangent to the ellipse x2/a2 +y2/b2 = 1, then c2 = a2m2 + b2.

=> ( p/sin α)2 = a2(cos α/sin α)2 + b2

=> p2/sin2α = a2 cos2α/sin2α + b2

=> p2 = a2 cos2α + b2 sin2α which is the required condition.

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