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If y=asinx+bcosx ,t h e ny^2+((dy)/(dx))...

If `y=asinx+bcosx ,t h e ny^2+((dy)/(dx))^2` is a function of `x` (b) function of`y` function of `xa n dy` (d) constant

A

function of x

B

function of y

C

function of x and y

D

constant

Text Solution

Verified by Experts

y=a sin x + b cos x
Differentiating with respect to x, we get
`(dy)/(dx)=a cos x - b sin x`
`"Now, "((dy)/(dx))^(2)=(a cos x - b sin x)^(2)`
`=a^(2)cos^(2)x+b^(2)sin^(2)x-2ab sin x cos x`
`"and "y^(2)=(a sin x + b cos x)^(2)`
`a^(2)sin^(2) x +b^(2)cos^(2) x +2ab sin x cos x`
`"So, "((dy)/(dx))^(2)+y^(2)=a^(2)(sin^(2)x + cos^(2)x)+b^(2)(sin^(2)x+ cos^(2)x)`
`=(a^(2)+b^(2))=` constant
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