If `y=asinx+bcosx` , then `y^2+((dy)/(dx))^2` is a

Options

A. function of x
B. function of y
C. function of x and y
D. constant

Answer

D

Explanation

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