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  • From any point on the hyperbola   <img alt="\mathrm{\frac{x^2}{a^2}-\frac{y^2}{b^2}=1}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Cfrac%7Bx%5E2%7D%7Ba%5E2%7D-%5Cf

From any point on the hyperbola   \mathrm{\frac{x^2}{a^2}-\frac{y^2}{b^2}=1} , tangents are 

drawn to the hyperbola  \mathrm{\frac{x^2}{a^2}-\frac{y^2}{b^2}=2} . The area cut-off by the     

chord of contact on the asymptotes is equal to  

Option: 1

a/2


Option: 2

ab


Option: 3

2ab


Option: 4

4ab


Answers (1)

best_answer

Let  \mathrm{P(X_1,Y_1)}  be a point on the hyperbola

\mathrm{\frac{x^2}{a^2}-\frac{y^2}{b^2}=1}

The chord of contact of tangents from P to the hyperbola

\mathrm{\frac{x^2}{a^2}-\frac{y^2}{b^2}=2}   is given by

\mathrm{\frac{x x_1}{a^2}-\frac{y y_1}{b^2}=2}                               ........(1)

The equations of the asymptotes are

               \mathrm{\frac{x }{a}-\frac{y }{b}=0}

and        \mathrm{\frac{x }{a}+\frac{y }{b}=0}

The points of intersection of (1) with the two asymptotes 

are given by

\mathrm{\begin{aligned} & x_1=\frac{2 a}{\left(x_1 / a\right)-\left(y_1 / b\right)}, y_1=\frac{2 b}{\left(x_1 / a\right)-\left(y_1 / b\right)} \\ & x_2=\frac{2 a}{\left(x_1 / a\right)+\left(y_1 / b\right)}, y_2=\frac{-2 b}{\left(x_1 / a\right)+\left(y_1 / b\right)} \end{aligned}}

Area of the said triangle     

\mathrm{\begin{aligned} & =\frac{1}{2}\left|x_1 y_2-x_2 y_1\right| \\ & =\frac{1}{2}\left|\frac{4 a b \times 2}{\left(x_1^2 / a^2\right)-\left(y_1^2 / b^2\right)}\right|=4 a b \end{aligned}}

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