The area of the triangle formed by the lines $x^{2} - y^{2} = 0$ and tangent at $(3, 2)$ to the hyperbola $x^{2} - y^{2} = 5$ is

9

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4

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5

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1

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Solution

The correct option is C $5$
Equation of tangent to the given hyperbola at $(3, 2)$ is $3 x - 2 y = 5$
and given lines are $y = x, y = - x$
Solving above equation we get intersection points which are $(0, 0), (5, 5), (1, - 1)$
Therefore, area of the triangle $= Δ = \frac{1}{2} ∣ ∣ ∣ ∣ \begin{matrix} 0 & 0 & 1 5 & 5 & 1 1 & - 1 & 1 \end{matrix} ∣ ∣ ∣ ∣ = \frac{1}{2} | (- 5 - 5) | = 5$
Hence, option 'C' is correct.

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