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

Options

A. `(-y)/x`
B. `x/y`
C. `y/x`
D. `(-x)/y`

Answer

A

Explanation

We have ,
`x = sqrt(a^(sin t)), y = sqrt(a^(cos-1)t)`
`therefore " " xy = sqrt(a^(sin-1t) . a^(cos-1t)) = sqrt(a^(sint+cos^(-1)t))`
`rArr xy = sqrt(a^(pi//2))" "(because sin^(-1) t+ cos^(-1) =(pi)/(2))`
On differentiating both sides w.r.t .x., we get
`x(dy)/(dx) + y =0`
`rArr " "(dy)/(dx) = (-y)/(x)`