Tangents are drawn to the hyperbola \(x^2-9 y^2=9\) from point \((3,2)\). Then, the area of the triangle…
Tangents are drawn to the hyperbola \(x^2-9 y^2=9\) from point \((3,2)\). Then, the area of the triangle formed by the tangents and the chord of contact is ____ sq units.
10
6
12
8
Solution
Given hyperbola \(\frac{x^2}{9}-\frac{y^2}{1}=1\)
So, \(y=m x \pm \sqrt{9 m^2-1}\) represent tangents which passes through \((3,2)\)
\(\Rightarrow \quad m=\frac{5}{12}\)
\(\therefore\) Tangents are
\(12 y=5 x+9 \text { and } x=3\)
Chord of contact will be \(\frac{x x_1}{a^2}-\frac{y y_1}{b^2}=1\)
\(\Rightarrow \quad \frac{3 x}{9}-\frac{2 y}{1}=1 \Rightarrow x-6 y=3\)
The vertices of the triangle formed with tangents and chord will be \((3,2),(3,0)\) and \(\left(-5,-\frac{4}{3}\right)\)
So, area of \(\Delta=\frac{1}{2}\left|\begin{array}{ccc}3 & 2 & 1 \\ 3 & 0 & 1 \\ -5 & -4 / 3 & 1\end{array}\right|=8\) unit \(^2\)