Let the hyperbola
\(H:\frac{x^2}{a^2}−y^2=1\)
and the ellipse
\(E:3x^2+4y^2=12\)
be such that the length of latus rectum of H is equal to the length of latus rectum of E. If eH and eE are the eccentricities of H and E respectively, then the value of
\(12 (e^{2}_H+e^{2}_E)\) is equal to _____ .
Let the hyperbola
\(H:\frac{x^2}{a^2}−y^2=1\)
and the ellipse
\(E:3x^2+4y^2=12\)
be such that the length of latus rectum of H is equal to the length of latus rectum of E. If eH and eE are the eccentricities of H and E respectively, then the value of
\(12 (e^{2}_H+e^{2}_E)\) is equal to _____ .
Correct Answer: 42
Solution and Explanation
The correct answer is 42
\(∴ H:\frac{x^2}{a^2}−\frac{y^2}{1}=1\)
∴ Length of latus rectum
\(=\frac{2}{a}\)
\(E:3x^2+4y^2=12\)
Length of latus rectum
\(= \frac{6}{2} = 3\)
\(\because \frac{2}{a} = 3 ⇒ a = \frac{2}{3}\)
\(12 (e^{2}_H+e^{2}_E)\)
\( = 12(1+\frac{9}{4})+(1-\frac{3}{4})\)
\(= 42\)
Top Questions on Hyperbola
- Let the foci of a hyperbola be \( (1, 14) \) and \( (1, -12) \). If it passes through the point \( (1, 6) \), then the length of its latus-rectum is:
- The foci of the hyperbola \[ 4x^2 - 9y^2 - 1 = 0 \] are:
- If \( e_1 \) and \( e_2 \) are respectively the eccentricities of the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) and its conjugate hyperbola, then the line \( \frac{x}{2e_1} + \frac{y}{2e_2} = 1 \) touches the circle having center at the origin, then its radius is:
- If the eccentricity of the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) is \( \sec \alpha \), then the area of the triangle formed by the asymptotes of the hyperbola with any of its tangent is:
- If \( \tanh x = \text{sech } y = \frac{3}{5} \) and \( e^{x+y} \) is an integer, then \( e^{x+y} \) is:
Questions Asked in JEE Main exam
- Consider ‘n’ is the number of lone pair of electrons present in the equatorial position of the most stable structure of \( \text{ClF}_3 \). The ions from the following with ‘n’ number of unpaired electrons are:
A. \( \text{V}^{3+} \)
B. \( \text{Ti}^{3+} \)
C. \( \text{Cu}^{2+} \)
D. \( \text{Ni}^{2+} \)
E. \( \text{Ti}^{2+} \)
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Concepts Used:
Hyperbola
Hyperbola is the locus of all the points in a plane such that the difference in their distances from two fixed points in the plane is constant.
Hyperbola is made up of two similar curves that resemble a parabola. Hyperbola has two fixed points which can be shown in the picture, are known as foci or focus. When we join the foci or focus using a line segment then its midpoint gives us centre. Hence, this line segment is known as the transverse axis.
