Question

If x sin (a + y) + sin a cos (a + y) = 0, prove that `"dy"/"dx" = (sin^2("a" + y))/sin"a"`

Answer 1

Given that: x sin (a + y) + sin a cos (a + y) = 0

⇒ x sin (a + y) = – sin a cos (a + y)

⇒ x = `(-sin"a" * cos("a" + y))/(sin ("a" + y))`

⇒ x = – sin a.cot (a + y)

Differentiating both sides w.r.t. y

⇒ `"dx"/"dy" = - sin"a"*"d"/"dy" cot("a" + y)`

⇒ `"dx"/"dy" = -sin"a"[-"cosec"^2("a" + y)`

⇒ `"dx"/"dy" = sin"a"/(sin^2("a" + y))`

∴ `"dy"/"dx" = 1/("dx"/"dy")`

= `1/(sin"a"/(sin^2("a" + y))`

Hence, `"dy"/"dx" = (sin^2("a" + y))/sin"a"`.

Hence proved.