Let y = y(x) be a function of x satisfying
y√(1 - x²) = k - x√(1 - y²) where k is a constant and
y( 1/2 ) = - 1/4. Then dy/dx at x = 1/2, is equal to :

Solution

Maths sir · 4 steps

Put x = sin a and y = sin b, and the whole equation becomes sin(a+b) = k.

1. Spot the pattern. Move the term to the left: y√(1-x²) + x√(1-y²) = k. Put x = sin a and y = sin b, so √(1-x²) = cos a and √(1-y²) = cos b. The left side becomes sin b cos a + sin a cos b = sin(a+b).
2. Differentiate. Since sin(a+b) = k, the sum a + b = sin⁻¹ x + sin⁻¹ y is a constant. Its derivative with respect to x is zero, which gives:

   1/√(1-x²) + 1/√(1-y²) dy/dx = 0 ⇒ dy/dx = -√(1-y²)/√(1-x²)

3. Put in numbers. At x = 1/2, y = -1/4: √(1-y²) = √(15/16) = √15/4 and √(1-x²) = √3/2. So dy/dx = -√15/4 × 2/√3 = -√5/2.
4. So. dy/dx at x = 1/2 is -√5/2, which is option (b).

The idea to keep. Whenever you see √(1-x²) and √(1-y²) together, try x = sin a, y = sin b. A sum formula usually appears and the derivative becomes one line.

Answer from the NTA final answer key. Solution by PrepWiser.