Question

Let y = y(x) be a function of x satisfying `ysqrt(1 - x^2) = k - xsqrt(1 - y^2)` where k is a constant and `y(1/2) = -1/4`. Then `(dy)/(dx)` at x = `1/2`, is equal to ______.

पर्याय

• `-sqrt(5)/4`

• `-sqrt(5)/2`

• `2/sqrt(5)`

• `sqrt(5)/2`

Answer 1

Let y = y(x) be a function of x satisfying `ysqrt(1 - x^2) = k - xsqrt(1 - y^2)` where k is a constant and `y(1/2) = -1/4`. Then `(dy)/(dx)` at x = `1/2`, is equal to `underlinebb(-sqrt(5)/2)`.

Explanation:

Given, x = `1/2`, y = `-1/4` `\implies` xy = `-1/8`

`y(1 - x^2) = k - xsqrt(1 - y^2)`

Differentiating both side w.r.t. x

`y.(1.(-2x))/(2sqrt(1 - x^2)) + y^'sqrt(1 - x^2)`

= `-{1.sqrt(1 - y^2) + (x.(-2y))/(2sqrt(1 - y^2))y^'}`

`\implies - (xy)/sqrt(1 - x^2) + y^'sqrt(1 - x^2) = -sqrt(1 - y^2) + (xy.y^')/sqrt(1 - y^2)`

`\implies y^'(sqrt(1 - x^2) - (xy)/sqrt(1 - y^2)) = (xy)/sqrt(1 - x^2) - sqrt(1 - y^2)`

`\implies y^'(sqrt(3)/2 + 1/(8. sqrt(15)/4)) = (-1)/(8.sqrt(3/2)) - sqrt(15)/4`

`\implies y^'((sqrt(45) + 1)/(2sqrt(15))) = -((1 + sqrt(45)))/(4sqrt(3))`

∴ y' = `-sqrt(5)/2`