InterviewSolution
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InterviewSolution
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From any point to the hyperbola `^2/a^2-y^2/b^2=1`, tangents are drawn to thehyperbola `x^2/a^2-y^2/b^2=2` The area cut off bythe chord of contact on the regionbetween the asymptotes is equal toA. `(ab)/(2)`B. `ab`C. `2ab`D. `4ab` |
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Answer» Let `P(x_(1),y_(1))` be a point on the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1`. Then, `(x_(1)^(2))/(a^(2))-(y_(1)^(2))/(b^(2))=1` The chord of contact of tangents from `P` to the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=2` is `(x x_(1))/(a^(2))-(y y_(1))/(b^(2))=2`..........`(i)` The equations of the asymptotes are `(x)/(a)-(y)/(b)=0` and `(x)/(a)+(y)/(b)=0` The points of intersection of `(i)` with the two asymptotes are given by `x_(1)=(2a)/((x_(1))/(a)-(y_(1))/(b))`, `y_(1)=(2b)/((x_(1))/(a)-(y_(1))/(b))`, `x_(2)=(2a)/((x_(1))/(a)+(y_(1))/(b))`, `y_(2)=(2b)/((x_(1))/(a)+(y_(1))/(b))`, `:.` Area of the triangle `=(1)/(2)(x_(1)y_(2)-x_(2)y_(1))=(1)/(2)((4abxx2)/((x_(1)^(2))/(a^(2))-(y_(1)^(2))/(b^(2))))=4ab` |
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