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If y=a sin x + b cos x, then ((dy)/dx)^...

If ` y=a sin x + b cos x`, then `((dy)/dx)^2+y^2` is

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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Knowledge Check

  • If y = a sin x + b cos x," then "y^(2) + ((dy)/(dx))^(2) =

    A
    `a^(2)-b^(2)`
    B
    `a^(2) +b^(2)`
    C
    `2a^(2)`
    D
    `2b^(2)`

  • If sin y=x cos ( a + y) , then (dy)/(dx) =

    A
    `(sin^2 (a+y))/(sin a)`
    B
    `( cos a)/( cos^2 (a +y))`
    C
    `(cos^2 (a+y))/(cos a)`
    D
    None

  • If sin^(2) x + cos^(2) y = 1 , " then " (dy)/(dx) is equal to

    A
    `(sin 2x)/(sin 2y)`
    B
    `(sin^(2)y)/(sin 2x)`
    C
    `(sin^(2) x)/(sin^(2)y)`
    D
    `(sin^(2) y)/(sin^(2) x)`

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    If sin^(2)x+2cos y+xy=0 then (dy)/(dx)=

    sin x(dy)/(dx)+y=y^(2)

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