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