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 65/1/1 · 3 marks · Short answer

CBSE Class 12 Mathematics 2024: question 27(a)

Set in the CBSE Class 12 board exam of 2024, paper 65/1/1, section C, question 27(a), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q27(a) · 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 65/1/1.

Hints

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

Hint 1

The terms $1 - x^{2}$ suggest the substitution $x = sin θ$ .

Hint 2

Use sum-to-product formulas for $cos θ + cos ϕ$ and $sin θ - sin ϕ$ .

Hint 3

Once $θ - ϕ$ equals a constant, rewrite in inverse sines and 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 $x = sin θ$ , $y = sin ϕ$ . Then the equation becomes $cos θ + cos ϕ = a (sin θ - sin ϕ)$ .
½ mark
$2 cos \frac{θ + ϕ}{2} cos \frac{θ - ϕ}{2} = 2 a cos \frac{θ + ϕ}{2} sin \frac{θ - ϕ}{2} \Rightarrow cot \frac{θ - ϕ}{2} = a$
½ mark
$θ - ϕ = 2 cot^{- 1} a \Rightarrow sin^{- 1} x - sin^{- 1} y = 2 cot^{- 1} a$
½ mark
Differentiating w.r.t. $x$ (RHS is constant): $\frac{1}{1 - x ^{2}} - \frac{1}{1 - y ^{2}} \frac{d y}{d x} = 0$
1 mark
$\frac{d y}{d x} = \frac{1 - y ^{2}}{1 - x ^{2}} = \frac{1 - y ^{2}}{1 - x ^{2}}$ . Hence proved.
½ mark

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

Where the marks are

Substitution $x = sin θ$ , $y = sin ϕ$ (0.5 mark)
Sum-to-product to get $cot \frac{θ - ϕ}{2} = a$ (0.5 mark)
Writes $sin^{- 1} x - sin^{- 1} y = 2 cot^{- 1} a$ (0.5 mark)
Differentiates correctly (1 mark)
Final result (0.5 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 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2024: question 27(a)

Hints

Step-by-step solution

Revise the chapter

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