Question

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

Options

• `(-y)/x`

• `x/y`

• `y/x`

• `(-x)/"y"`

Answer 1

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

Explanation:

We have,

x = `sqrt("a"^(sin^-1 "t")), "y" = sqrt("a"^(cos^-1 "t"))`

∴ xy = `sqrt("a"^(sin^-1"t") * "a"^(cos^-1 "t"))`

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

`=> "xy" = sqrt("a"^(pi/2))  ...(because sin^-1 "t" + cos^-1 "t" = pi/2)`

On differentiating both sides w.r.t. x, we get

`x "dy"/"dx" + "y" = 0`

`=> "dy"/"dx" = (- "y")/x`