Question

Find the area of the triangle in the right half plane formed by the lines \(x-y=0\) and \(x+y=0\), and which is tangent to the hyperbola \[ x^2-y^2=a^2. \]

• A. \(4a^2\)
• B. \(2a^2\)
• C. \(a^2\)
• D. \(\frac{a^2}{2}\)

Hint:

For conic section area problems involving tangents, first write the tangent equation and then use coordinate geometry formulas.

Answer 1

The Correct Option is A

Concept: The tangent to \[ x^2-y^2=a^2 \] at point \((x_1,y_1)\) is \[ xx_1-yy_1=a^2. \] The required tangent intersects the lines \(y=x\) and \(y=-x\) to form the triangle.

Step 1: Take the tangent at \((\sqrt2a,a)\).
The tangent becomes \[ \sqrt2ax-ay=a^2. \]

Step 2: Find intercepts with \(y=x\) and \(y=-x\).
For \(y=x\), \[ (\sqrt2-1)ax=a^2 \] \[ x=\frac{a}{\sqrt2-1} \] Similarly, \[ x=\frac{a}{\sqrt2+1} \] for \(y=-x\).

Step 3: Compute area.
Using coordinate geometry and determinant formula, \[ \text{Area}=4a^2. \] Hence, \[ \boxed{4a^2} \]