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Question
If `ysqrt(1 - x^2) + xsqrt(1 - y^2)` = 1, then prove that `(dy)/(dx) = - sqrt((1 - y^2)/(1 - x^2))`
Solution
`ysqrt(1 - x^2) + xsqrt(1 - y^2)` = 1
Let sin–1x = A and sin–1y = B.
Then x = sinA and y = sinB
`ysqrt(1 - x^2) + xsqrt(1 - y^2)` = 1
⇒ sinBcosA + sinAcosB = 1
⇒ sin(A + B) = 1
⇒ A + B = sin–11 = `pi/2`
⇒ sin–1x + sin–1y = `pi/2`
Differentiating w.r.to x, we obtain `(dy)/(dx) = - sqrt((1 - y^2)/(1 - x^2))`
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