AP EAMCET · Maths · Hyperbola
The locus of the point of intersection of the tangents drawn at the extremities of a normal chord of the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) is
- A \(\frac{a^2}{x^2}-\frac{b^2}{y^2}=a^2+b^2\)
- B \(\frac{a^4}{x^2}-\frac{b^4}{y^2}=\left(a^2-b^2\right)^2\)
- C \(\frac{a^3}{x^2}-\frac{b^3}{y^2}=\left(a^2+b^2\right)^2\)
- D \(\frac{a^6}{x^2}-\frac{b^6}{y^2}=\left(a^2+b^2\right)^2\)
Answer & Solution
Correct Answer
(D) \(\frac{a^6}{x^2}-\frac{b^6}{y^2}=\left(a^2+b^2\right)^2\)
Step-by-step Solution
Detailed explanation
Let the point of intersection be \( (h, k) \). The chord of contact is \( \frac{xh}{a^2}-\frac{yk}{b^2}=1 \). The normal at \( (x_1, y_1) \) on the hyperbola is \( \frac{a^2x}{x_1}+\frac{b^2y}{y_1}=a^2+b^2 \). Comparing the chord of contact with the normal (as it's a normal…
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