InterviewSolution
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InterviewSolution
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If`sqrt(1-x^2)+sqrt(1-y^2)=a(x-y),"p r o v et h a t"(dy)/(dx)sqrt((1-y^2)/(1-x^2))` |
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Answer» Given: `sqrt(1-x^(2))+sqrt(1-y^(2))=a(x-y)." …(i)"` Putting `x=sin theta and y = sin phi`, it becomes `cos theta + cos phi =a(sin theta - sin phi)` `rArr(cos theta+cos phi)/(sin theta-sin phi)=a` `rArr(2cos((theta+phi)/(2))cos((theta-phi)/(2)))/(2cos((theta+phi)/(2))sin((theta-phi)/(2)))=a` `rArr cot((theta-phi)/(2))=a rArr theta-phi = 2 cot^(-1)a` `rArr sin^(-1)x sin^(-1)y=2 cot^(-1)a." ...(ii)"` On differentiating both sides of (ii) w.r.t. x, we get `(1)/(sqrt(1-x^(2)))-(1)/(sqrt(1-y^(2))).(dy)/(dx)=0` Hence, `(dy)/(dx)=sqrt((1-y^(2))/(1-x^(2))).` |
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