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Question: If x sin (α + y) = sin y and sec<sup>2</sup>y\(\frac{dy}{dx}\) = \(\frac{m}{(x^{2} + 2nx + 1)}\). Th...

If x sin (α + y) = sin y and sec2ydydx\frac{dy}{dx} = m(x2+2nx+1)\frac{m}{(x^{2} + 2nx + 1)}. Then

A

m – n = 1

B

m + n = 1

C

m2 + n2= 1

D

m = n

Answer

m2 + n2= 1

Explanation

Solution

x sin α cos y + x cos α sin y = sin y

x sin α cot y + x cos α = 1

cot y = 1−xcos⁡αxsin⁡α\frac{1 - x\cos\alpha}{x\sin\alpha}

tan y = xsin⁡α1−xcos⁡α\frac{x\sin\alpha}{1 - x\cos\alpha}

sec2y . dydx\frac{dy}{dx}

= sin⁡α(1−xcos⁡α)−xsin⁡α(−cos⁡α)(1−xcos⁡α)2\frac{\sin\alpha(1 - x\cos\alpha) - x\sin\alpha( - \cos\alpha)}{(1 - x\cos\alpha)^{2}}sec2y . dydx\frac{dy}{dx}

= sin⁡α−xsin⁡αcos⁡α+xsin⁡αcos⁡α(1−xcos⁡α)2\frac{\sin\alpha - x\sin\alpha\cos\alpha + x\sin\alpha\cos\alpha}{(1 - x\cos\alpha)^{2}}

sec2y dydx\frac{dy}{dx} = sin⁡α(1−xcos⁡α)2\frac{\sin\alpha}{(1 - x\cos\alpha)^{2}}

m = sin α n = – cos α

m2 + n2 = 1