Question:
If the eccentricity of a hyperbola is $\sqrt{3}$, then the eccentricity of its conjugate hyperbola is:
If the eccentricity of a hyperbola is $\sqrt{3}$, then the eccentricity of its conjugate hyperbola is:
Show Hint
The elegant relationship $\frac{1}{e_1^2} + \frac{1}{e_2^2} = 1$ is incredibly powerful and appears frequently in entrance exams. Memorize it!
Updated On: May 31, 2026
- $\sqrt{\frac{3}{2}}$
- $\sqrt{3}$
- $\frac{\sqrt{3}}{2}$
- $\sqrt{2}$
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Step 1: Concept
If $e_1$ and $e_2$ are the eccentricities of a hyperbola and its conjugate hyperbola, they satisfy the reciprocal identity: \[ \frac{1}{e_1^2} + \frac{1}{e_2^2} = 1 \]
Step 2: Meaning
We are given $e_1 = \sqrt{3}$ and need to solve for $e_2$.
Step 3: Analysis
Substituting $e_1 = \sqrt{3}$ into the identity: \[ \frac{1}{(\sqrt{3})^2} + \frac{1}{e_2^2} = 1 \implies \frac{1}{3} + \frac{1}{e_2^2} = 1 \] \[ \frac{1}{e_2^2} = 1 - \frac{1}{3} = \frac{2}{3} \] \[ e_2^2 = \frac{3}{2} \implies e_2 = \sqrt{\frac{3}{2}} \]
Step 4: Conclusion
The eccentricity of the conjugate hyperbola is $\sqrt{\frac{3}{2}}$. Final Answer: (A)
Was this answer helpful?
0
0
Top AP EAPCET Mathematics Questions
- $D = \left\{x\in\mathbb{R}: f\left(x\right) =\sqrt{\frac{x - \left|x\right|}{x - \left[x\right]}} \text{is defined} \right\}$ and $C$ be the range of the real function $g(x) = \frac{2x}{4 + x^{2}}$. Then $D \cap C$
- $f\left(x\right) = \frac{x}{e^{x}-1} + \frac{x}{2} + 2 \cos^{3} \frac{x}{2} $ on $R-\left\{0\right\} $ is
- Consider the following lists.
List-I List-II (A) $f(x) = \frac{|x+2|}{x+2} , x \ne -2 $ (I) $[\frac{1}{3} , 1 ]$ (B) $(x)=|[x]|,x \in [R$ (II) Z (C) $h(x) = |x - [x]| , x \in [R$ (III) W (D) $f(x) = \frac{1}{2 - \sin 3x} , x \in [R$ (IV) [0, 1) (V) { -1, 1} - If $[x]$ is the greatest integer less than or equal to $x$ and $|x|$ is the modulus of $x$. then the system of three equations $2x + 3 | y | + 5[z] = 0, x + |y| - 2[z] = 4, x + |y| + |z| = 1$ has
- Investigate the values of $\lambda$ and $\mu$ for the system $x+2 y+3 z=6, x+3 y+5 z=9$, $2 x+5 y+\lambda z=\mu$ and match the values in List - I with the items in List - II.
List I List II (A) $\lambda=8, \mu \neq 15$ 1. Infinitely many solutions (B) $\lambda \neq 8, \mu \in R$ 2. No solution (C) $\lambda=8, \mu=15$ 3. Unique solution
View More Questions
Top AP EAPCET sections of a cone Questions
- The equation of the parabola with focus at $(3, 0)$ and directrix $x + 3 = 0$ is:
- If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12, then the length of its latus rectum is:
- The line $y = mx + c$ touches the ellipse $9x^2 + 16y^2 = 144$ if the value of $c^2$ is:
- The equation of the tangent to the parabola $y^2 = 8x$ which is parallel to the line $2x - y + 5 = 0$ is:
- The equation of the parabola with focus at $(3, 0)$ and directrix $x + 3 = 0$ is:
View More Questions
Top AP EAPCET Questions
- A body is projected from the ground at an angle of $\tan^{-1} \left( \frac{8}{7}\right)$ with the horizontal. The ratio of the maximum height attained by it to its range is
- The four quantum numbers of the valence electron of potassium are :
- A lens forms real and virtual images of an object, when the object is at $u_{1}$ and $u_{2}$ distances respectively. If the size of the virtual image is double that of the real image, then the focal length of the lens is (take, the magnification of the real image as $m$ )
- Two spheres $P$ and $Q$, each of mass $200\, g$ are attached to a string of length one metre as shown in the figure. The string and the spheres are then whirled in a horizontal circle about $O$ at a constant angular speed. The ratio of the tension in the string between $P$ and $Q$ to that of between $P$ and $O$ is $(P$ is at mid-point of the line joining $O$ and $Q$ )
- The volume of $0.02\, M$ acidified permanganate solution required for complete reaction of $60\, mL$ of $0.01\, MI^{-}$ ion solution to form $I_2$ in $mL$ is
View More Questions