AP EAMCET · Maths · Hyperbola
If \(S \equiv \frac{\mathrm{X}^2}{\mathrm{k}-7}+\frac{\mathrm{y}^2}{11-\mathrm{k}}-1=0, \mathrm{k} \in \mathrm{R}-\{7,11\}\), then which one of the following statements is incorrect?
- A \(\mathrm{S}=0\) represents a circle with radius \(\sqrt{2}\), when \(\mathrm{k}=9\)
- B \(S=0\) represents an ellipse with eccentricity \(\sqrt{\frac{2}{3}}\), when \(\mathrm{k}=10\)
- C \(S=0\) represents a hyperbola with eccentricity \(\sqrt{\frac{6}{5}}\) when \(\mathrm{k}=12\)
- D \(S=0\) represents a hyperbola with eccentricity \(\sqrt{\frac{3}{2}}\) when \(\mathrm{k}=13\)
Answer & Solution
Correct Answer
(C) \(S=0\) represents a hyperbola with eccentricity \(\sqrt{\frac{6}{5}}\) when \(\mathrm{k}=12\)
Step-by-step Solution
Detailed explanation
Given that \(S=\frac{x^2}{k-7}+\frac{y^2}{11-k}-1=0...(i)\) Put \(k=13\) \(\Rightarrow \frac{x^2}{6}-\frac{y^2}{2}-1=0 \Rightarrow \frac{x^2}{6}-\frac{y^2}{2}=1\) Represent equation of hyperbola, where \(a^2=6\) and \(b^2=2\). Now,…
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