InterviewSolution
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InterviewSolution
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If `x = a sin 2t(1 + cos 2t) and y = b cos 2 t(1 – cos 2 t),` show that `((d y)/(d x))_(at t =pi/4)=b/a.` |
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Answer» We have `x=a sin 2t(1+cos 2t)` `rArr(dx)/(dt)=a.[sin2t(-2 sin 2t)+(1+cos 2t)(2cos 2t)]` `=(2a).[-sin^(2)2t+cos 2t+cos^(2)2t]` `=(2a)[cos4t+cos2t]" "{because" "(cos^(2)2t-sin^(2)2t)=cos4t}.` And, `y=b cos 2t (1-cos2t)` `rArr (dy)/(dx)=b[cos2t(2sin 2t)+(1-cos2t)(-2sin2t)]` `=(2b)[sin 2t cos 2t-sin 2t +sin 2t cos 2t)]` `(2b)[2sin 2t cos 2t -sin 2t]=(2b)[sin 4t-sin 2t].` `therefore(dy)/(dx)=((dy//dt))/((dx//dt))=((2b)(sin 4t -sin 2t))/((2a)(cos 4t +cos2t))` `rArr((dy)/(dx))_((t=(pi)/4))=(b)/(a).({sin(4xx(pi)/(4))-sin(2xx(pi)/(4))})/({cos(4xx(pi)/(4))+cos(2xx(pi)/(4))})` `=(b)/(a).((sin pi-sin.(pi)/(2)))/((cos pi+cos.(pi)/(2)))=(b)/(a).((0-1))/((-1+0))=(b)/(a).` |
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