Question

If x = `sqrt(a^(sin^-1 t)` and y = `sqrt(a^(cos^-1 t)`, then `dy/dx` = ______.

Options

• `x/y`

• `y/x`

• `(-y)/x`

• `(-x)/y`

Answer 1

If x = `sqrt(a^(sin^-1 t)` and y = `sqrt(a^(cos^-1 t)`, then `dy/dx` = `underlinebb((-y)/x)`.

Explanation:

We have,

x = `sqrt(a^(sin^-1 t)`   ...(i)

and y = `sqrt(a^(cos^-1 t)`  ...(ii)

On multiplying equations (i) and (ii) we get

xy = `sqrt(a^(sin^-1 t) . a^(cos^-1 t)`

= `sqrt(a^(sin^-1 t) + a^(cos^-1 t)`

`\implies` xy = `sqrt(a^(π//2))`  ...`[∵ sin^-1 x + cos^-1 x = π/2]`

`\implies` xy = `a^(π//4)`  ...(iii)

On differentiating both sides of equation (iii) w.r.t x, we get

`x dy/dx + y` = 0

`\implies dy/dx = (-y)/x`