<p>If the normal to the rectangular hyperbola \(x^2 - y^2 = 4\) at a point P meets the coordinate axes in Q and R and O is the centre of the hyperbola, then:</p>
Step-by-Step Solution
Key Concept: Use parametric form of the rectangular hyperbola and properties of normals to establish distance relationships.
<p>Let P be a point \((2\sec\theta, 2\tan\theta)\) on the hyperbola \(x^2 - y^2 = 4\). The tangent at P has slope \(\frac{\tan\theta}{\sec\theta} = \sin\theta\). The normal has slope \(-\cot\theta\). The normal equation is \(y - 2\tan\theta = -\cot\theta(x - 2\sec\theta)\). Setting \(x = 0\) gives Q on the y-axis, and setting \(y = 0\) gives R on the x-axis. By calculation, \(PQ = PR\) (P is equidistant from both axes intersections) and \(QR = 2OP\) (by the geometry of the configuration).</p>
Correct Answer: c, d