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

dy/dx=√((1-y²)/(1-x²)) (proved)

MathsCBSE Class 12 · Board exam 2024, set 65/5/1 · 3 marks · Short answer

CBSE Class 12 Mathematics 2024: question 26(b)

Set in the CBSE Class 12 board exam of 2024, paper 65/5/1, section C, question 26(b), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q26(b) · 3 marks · Short answerContinuity and Differentiability

If $1 - x^{2} + 1 - y^{2} = a (x - y)$ , 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 2024 board paper, set 65/5/1.

Hints

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

Hint 1

Try the substitution $x = sin A$ , $y = sin B$ .

Hint 2

Use $cos A + cos B$ and $sin A - sin B$ sum-to-product identities, then cancel $cos \frac{A + B}{2}$ .

Hint 3

The constant $2 cot^{- 1} a$ differentiates to 0.

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 $x = sin A$ , $y = sin B$ , so $A = sin^{- 1} x$ , $B = sin^{- 1} y$ . The equation becomes $cos A + cos B = a (sin A - sin B)$ .
½ mark
$2 cos \frac{A + B}{2} cos \frac{A - B}{2} = 2 a cos \frac{A + B}{2} sin \frac{A - B}{2}$ .
1 mark
$cot \frac{A - B}{2} = a \Rightarrow A - B = 2 cot^{- 1} a$ , i.e. $sin^{- 1} x - sin^{- 1} y = 2 cot^{- 1} a$ .
½ mark
Differentiating w.r.t. 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.
1 mark

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

Where the marks are

Substitution $x = sin A$ , $y = sin B$ (½ mark)
Applies sum-to-product formulas (1 mark)
Obtains $sin^{- 1} x - sin^{- 1} y = 2 cot^{- 1} a$ (½ mark)
Differentiates and gets the result (1 mark)

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 paper2024 board paper, set 65/5/1: every question, by section.

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CBSE Class 12 Mathematics 2024: question 26(b)

Hints

Step-by-step solution

Revise the chapter

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