InterviewSolution
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InterviewSolution
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If `x=a(1-costheta)`, `y=a(theta+sintheta)`, prove that `(d^2y)/(dx^2)=-1/a`at `theta=pi/2`. |
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Answer» We have `x=a(theta+sin theta)and y=a(1-cos theta)` `rArr (dx)/(d theta)=a(1+cos theta) and (dy)/(d theta)=a sin theta` `rArr(dy)/(dx)=((dy//d theta))/((dx//d theta))` `=(asin theta)/(a(1+costheta))=(2sin(theta//2)cos(theta//2))/(2cos^(2)(theta//2))=tan.(theta)/(2)` `rArr(d^(2)y)/(dx^(2))=(d)/(dx)(tan.(theta)/(2))=(1)/(2)sec^(2).(theta)/(2).(d theta)/(dx)=((1)/(2)sec^(2).(theta)/(2))xx(1)/(a(1+cos theta))` `rArr((d^(2)y)/(dx^(2)))_(theta=(pi)/(2))=(1)/(2)sec^(2).(pi)/(4).(1)/(a(1+cos.(pi)/(2)))=(1)/(a).` |
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