InterviewSolution
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InterviewSolution
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Find `(dy)/(dx)`for `y=tan^(-1)sqrt((a-x)/(a+x),)-a |
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Answer» `y=tan^(-1){sqrt((a-x)/(a+x))}," where " -a lt x lt a` Substituting `x= a cos theta,` we get `y=tan^(-1){sqrt((a-a cos theta)/(a+a cos theta))},` `=tan^(-1){sqrt((1- cos theta)/(1+ cos theta))},` `=tan^(-1){sqrt(tan^(2)""(theta)/(2))}` `=tan^(-1)|tan""(theta)/(2)|` Also, for `-a lt x lt a, -1 lt cos theta lt 1` `"or "theta in (0,pi) or (theta)/(2) in (0, (pi)/(2))` `therefore" "y=tan^(-1)|tan""(theta)/(2)|=tan^(-1)(tan""(theta)/(2))` `=(theta)/(2)=(1)/(2) cos^(-1) ((x)/(a))` `"or "(dy)/(dx)=-(1)/(2)xx(1)/(sqrt(1-(x^(2))/(a^(2))))(d)/(dx)((x)/(a))=-(1)/(2sqrt(a^(2)-x^(2)))` |
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