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  • A tangent is drawn at the vertex of the hyperbola  <img alt="\mathrm{\f

A tangent is drawn at the vertex \mathrm{A(a, 0)} of the hyperbola  \mathrm{\frac{x^2}{a^2}-\frac{y^2}{b^2}=1.} The area formed by the asymptotes and this tangent is

Option: 1

ab


Option: 2

2ab


Option: 3

4ab


Option: 4

8ab


Answers (1)

best_answer

The equations of asymptotes are

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

And tangent at \mathrm{A(a, 0) \, \, is\, \, x=a}

Hence, P is \mathrm{(a, b), Q} is \mathrm{(a,-b)} and O is \mathrm{(0,0)}
\mathrm{ \text { Area of } \begin{aligned} \triangle O P Q & = \pm \frac{1}{2}\left|\begin{array}{ccc} a & b & 1 \\ a & -b & 1 \\ 0 & 0 & 1 \end{array}\right| \\ & =a b \end{aligned} }

The answer is (a) 

Posted by

Sanket Gandhi

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