If the line xcosα+ysinα=p is a tangent to the ellipse a2x2+b2y2=1, then prove that a2cos2α+b2sin2α=p2

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Question

If the line $x cos α + y sin α = p$ is a tangent to the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ , then prove that $a^{2} cos^{2} α + b^{2} sin^{2} α = p^{2}$

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Michael Discussed

If the line

x cos α + y sin α = p

is a tangent to the ellipse

\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1

, then prove that

a^{2} cos^{2} α + b^{2} sin^{2} α = p^{2}

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Question Text

If the line

x cos α + y sin α = p

is a tangent to the ellipse

\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1

, then prove that

a^{2} cos^{2} α + b^{2} sin^{2} α = p^{2}