If $\sqrt{1-x^{2}}+\sqrt{1-y^{2}}=a(x-y)$, prove that $\dfrac{dy}{dx}=\sqrt{\dfrac{1-y^{2}}{1-x^{2}}}$.

$\sqrt{\dfrac{1-y^{2}}{1-x^{2}}}$ (proved)

CBSE Class 12 · DifferentiationShort Answer

1 - x^{2} + 1 - y^{2} = a (x - y)

, prove that

\frac{d y}{d x} = \frac{1 - y ^{2}}{1 - x ^{2}}

Answer: $\frac{1 - y ^{2}}{1 - x ^{2}}$ (proved)

Put

x = sin α

and

y = sin β

, so

1 - x^{2} = cos α, 1 - y^{2} = cos β

.
The relation becomes:

cos α + cos β = a (sin α - sin β)

Use sum-to-product on both sides:

2 cos \frac{α + β}{2} cos \frac{α - β}{2} = a \cdot 2 cos \frac{α + β}{2} sin \frac{α - β}{2}

Cancel

2 cos \frac{α + β}{2}

cot \frac{α - β}{2} = a \Rightarrow α - β = 2 cot^{- 1} a = constant

Hence

sin^{- 1} x - sin^{- 1} y = constant

. Differentiate w.r.t.

x

\frac{1}{1 - x ^{2}} - \frac{1}{1 - y ^{2}} \frac{d y}{d x} = 0

\frac{d y}{d x} = \frac{1 - y ^{2}}{1 - x ^{2}}

CBSE Class 12 Board · Subjective