AP EAMCET · Maths · Hyperbola

The locus of the point of intersection of the tangents at the
endpoints of normal chords of the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) is

A \(\frac{a^6}{x^2}+\frac{b^6}{y^2}=\left(a^2+b^2\right)^2\)
B \(\frac{a^6}{x^2}-\frac{b^6}{y^2}=\left(a^2+b^2\right)^2\)
C \(\frac{a^6}{x^2}-\frac{b^6}{y^2}=\left(a^2-b^2\right)^2\)
D \(\frac{a^6}{x^2}+\frac{b^6}{y^2}=\left(a^2-b^2\right)^2\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(\frac{a^6}{x^2}-\frac{b^6}{y^2}=\left(a^2+b^2\right)^2\)

Step-by-step Solution

Detailed explanation

Let \(p(h, k)\) be point of intersection of tangents at end points of normal chord Standard equation of normal chord of hyperbola \(a x \cos \theta+b y \cot \theta=a^2+b^2\) ...(i) for \(p(h, k)\), equation of chord of contact of tangents to hyperbola 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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The locus of the point of intersection of the tangents at theendpoints of normal chords of the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) is

Step-by-step Solution

The locus of the point of intersection of the tangents at the
endpoints of normal chords of the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) is