JEE Mains · Maths · STD 11 - 10.2 parabola,ellipse,hyperbola
Let the hyperbola \(H : \frac{ x ^{2}}{ a ^{2}}- y ^{2}=1\) and the ellipse \(E: 3 x^{2}+4 y^{2}=12\) be such that the length of latus rectum of \(H\) is equal to the length of latus rectum of \(E\). If \(e_{ H }\) and \(e_{ E }\) are the eccentricities of \(H\) and \(E\) respectively, then the value of \(12\left( e _{ H }^{2}+ e _{ E }^{2}\right)\) is equal to
- A \(42\)
- B \(40\)
- C \(36\)
- D \(47\)
Answer & Solution
Correct Answer
(A) \(42\)
Step-by-step Solution
Detailed explanation
\(\frac{ x ^{2}}{ a ^{2}}-\frac{ y ^{2}}{1}=1\) \(e _{ H }=\sqrt{1+\frac{1}{ a ^{2}}} \quad \frac{ x ^{2}}{4}+\frac{ y ^{2}}{3}=1\) \(\ell \cdot R .=\frac{2}{ a } \quad \ell R =\frac{2 \times 3}{\frac{2}{1-\frac{3}{4}}}=\frac{1}{2}\) \(\frac{2}{ a }=3\) \(a =\frac{2}{3}\)…
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