If √(1-x²)+√(1-y²)=a(x-y), then prove that dy/dx=√((1-y²)/(1-x²)).

Proved: dy/dx = √((1-y²)/(1-x²))

MathsCBSE Class 12 · Board exam 2025, set 65/1/1 · 5 marks · Long answer

CBSE Class 12 Mathematics 2025: question 32(a)

Set in the CBSE Class 12 board exam of 2025, paper 65/1/1, section D, question 32(a), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q32(a) · 5 marks · Long answerContinuity and Differentiability

If $1 - x^{2} + 1 - y^{2} = a (x - y)$ , then prove that $\frac{d y}{d x} = \frac{1 - y ^{2}}{1 - x ^{2}}$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 65/1/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

The terms $1 - x^{2}$ suggest the substitution $x = sin A$ , $y = sin B$ .

Hint 2

Use the identities for $cos A + cos B$ and $sin A - sin B$ , then cancel the common factor.

Hint 3

You should get $sin^{- 1} x - sin^{- 1} y$ equal to a constant. Then differentiate.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Put $x = sin A$ and $y = sin B$ , so $A = sin^{- 1} x$ and $B = sin^{- 1} y$ . Then $1 - x^{2} = cos A$ and $1 - y^{2} = cos B$ .
1 mark
The equation becomes $cos A + cos B = a (sin A - sin B)$ . Using sum-to-product formulas: $2 cos \frac{A + B}{2} cos \frac{A - B}{2} = 2 a cos \frac{A + B}{2} sin \frac{A - B}{2}$ .
1 mark
Cancel $2 cos \frac{A + B}{2}$ : $cot \frac{A - B}{2} = a \Rightarrow A - B = 2 cot^{- 1} a$ .
1 mark
So $sin^{- 1} x - sin^{- 1} y = 2 cot^{- 1} a$ , which is a constant.
½ mark
Differentiate both sides with respect to $x$ : $\frac{1}{1 - x ^{2}} - \frac{1}{1 - y ^{2}} \frac{d y}{d x} = 0 \Rightarrow \frac{d y}{d x} = \frac{1 - y ^{2}}{1 - x ^{2}} = \frac{1 - y ^{2}}{1 - x ^{2}}$ . Hence proved.
1½ marks

Answer: Proved: $\frac{d y}{d x} = \frac{1 - y ^{2}}{1 - x ^{2}}$

Where the marks are

Substitutes $x = sin A$ , $y = sin B$ (1 mark)
Applies sum-to-product formulas correctly (1 mark)
Gets $cot \frac{A - B}{2} = a$ , i.e. $A - B = 2 cot^{- 1} a$ (1 mark)
Rewrites as $sin^{- 1} x - sin^{- 1} y = 2 cot^{- 1} a$ (½ mark)
Differentiates to reach the required result (1½ marks)

Revise the chapter

Chapter: Continuity and Differentiability (notes, formulas, where marks are lost and every past question)
Topic: Implicit Differentiation

Practise this chapterBoard-style questions from Continuity and Differentiability, one at a time, marked against the answer key and marking points.Back to the full paper2025 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2025: question 32(a)

Hints

Step-by-step solution

Revise the chapter

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