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If x=sectheta-costheta,y=sec^(10)theta-c...

If `x=sectheta-costheta,y=sec^(10)theta-cos^(10)theta` and `(x^(2)+4)((dy)/(dx))^(2)=k(y^(2)+4)`. Then k is equal to

A

`(1)/(100)`

B

1

C

10

D

100

Text Solution

Verified by Experts

The correct Answer is:
D
`because x^(2) + 4 = (sec theta - cos theta )^(2) + 4 = (sec theta + cos theta )^(2) `
Similarly , ` x^(2) + 4 = (sec^(10) theta + cos^(10) theta )^(2)`
Now , ` (dx)/(d theta ) = sec theta tan theta + sin theta = tan theta (sec theta + cos theta )`
and ` (dy)/(dx) = 10 sec^(9) theta sec theta tan theta - 10 cos^(9) theta (-sin theta)`
`= 10 tan theta (sec ^(10) theta + cos ^(10) theta )`
or ` (x^(2) + 4) ((dy)/(dx))^(2) = 100 (y^(2) + 4)`
Hence , ` k = 100
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