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Question

If a curve is represented parametrically by x=sin(t+7π12)+sin(tπ12)+sin(t+3π12), y=cos(t+7π12)+cos(tπ12)+cos(t+3π12), then the value of ddt(xyyx) at t=π8 is

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Solution

Given :
x=sin(t+7π12)+sin(tπ12)+sin(t+3π12)x=2sin(t+π4)cos(π3)+sin(t+π4)x=2sin(t+π4)
Similarly
y=cos(t+7π12)+cos(tπ12)+cos(t+3π12)y=2cos(t+π4)cos(π3)+cos(t+π4)y=2cos(t+π4)

Now,
xy=tan(t+π4)yx=cot(t+π4)
Therefore,
ddt(xyyx)=sec2(t+π4)+cosec2(t+π4)ddt(xyyx)=1cos2(t+π4)sin2(t+π4)ddt(xyyx)=4sin2(2t+π2)
Putting t=π8, we get
ddt(xyyx)=8

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