TS EAMCET · Maths · Hyperbola
If the product of the slopes of the tangents drawn from an external point \(P\) to the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) is a constant \(k^2\), then the locus of \(P\) is
- A \(y^2+b^2=k^2\left(x^2-a^2\right)\)
- B \(y^2-b^2=k^2\left(x^2-a^2\right)\)
- C \(x^2+b^2=k^2\left(y^2-a^2\right)\)
- D \(x^2-b^2=k^2\left(y^2-a^2\right)\)
Answer & Solution
Correct Answer
(A) \(y^2+b^2=k^2\left(x^2-a^2\right)\)
Step-by-step Solution
Detailed explanation
Let the point be \(P(h, k)\). Tangent from \((h, k)\) to hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) is \(\left(\frac{h x}{a^2}-\frac{k y}{b^2}-1\right)^2=\left(\frac{h^2}{a^2}-\frac{k^2}{b^2}-1\right)\) \[ \left(\frac{x^2}{a^2}-\frac{y^2}{b^2}-1\right) \]…
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