Hyperbola
Normal Properties
Grade 11
Question:
<p>If the normal to the hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) at any point \(P(a\sec\theta, b\tan\theta)\) meets the transverse and conjugate axes in G and g respectively and if F is the foot of perpendicular to the normal at P from the centre C, then the value of \(|PF| \cdot |PG|\) is equal to:</p>
<p>(a) \(b^2(\sec^2\theta + \tan^2\theta)\)</p>
<p>(b) \(a^2\)</p>
<p>(c) \(b^2\)</p>
<p>(d) \(b^2\sec^2\theta + a^2\tan^2\theta\)</p>
Step-by-Step Solution
Key Concept: The product of the distances from a point on the hyperbola to the foot of perpendicular from the centre and to the intersection point on the transverse axis is a constant.
<p>The product \(|PF| \cdot |PG|\) can be calculated by multiplying the expressions for PF and PG found in the previous parts. After simplification, the result is a constant \(b^2\), independent of \(\theta\).</p>
Correct Answer: C