The locus of a variable point whose chord of contact w.r.t. the hyperbola x2a2-y2b2=1 subtends a right angle at the origin is

AP EAMCET · Maths · Hyperbola

The locus of a variable point whose chord of contact w.r.t. the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ subtends a right angle at the origin is

A $\frac{x^{2}}{4 a^{2}} - \frac{y^{2}}{4 b^{2}} = 1$
B $(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}}) = \frac{x^{2}}{a^{4}} + \frac{y^{2}}{b^{4}}$
C $\frac{x}{a} - \frac{y}{b} = \frac{1}{a^{2}} + \frac{1}{b^{2}}$
D $\frac{x^{2}}{a^{4}} + \frac{y^{2}}{b^{4}} = \frac{1}{a^{2}} - \frac{1}{b^{2}}$

Verified Solution

Answer & Solution

Correct Answer

(D) $\frac{x^{2}}{a^{4}} + \frac{y^{2}}{b^{4}} = \frac{1}{a^{2}} - \frac{1}{b^{2}}$

Step-by-step Solution

Detailed explanation

Given, The equation of the given hyperbola is x2a2-y2b2=1 ⋯1 Now let h,k be the pole of a chord PQ of the hyperbola Then the straight line PQ is the polar of the point h,k w.r.t. the hyperbola and so its equation is…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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