New chat PYQ Library Doubts Rank report

Sign in to see Recents

Your guest activity stays on this device

Sign in to save progress →

Hyperbola question

2016 · Shift 0 · Q31

Guest · filters and generic practice availableBrowsing as a guest · PYQ filters and generic practice are available. Sign in only for personalised features and saved progress.

PYQ Library
JEE Main
Mathematics
Hyperbola
2016 · Shift 0 · Q31

Hyperbola question

2016 · Shift 0 · Q31

JEE MainMathematicsHyperbolaMCQ+4 / −1

The eccentricity of the hyperbola whose length of the latus rectum is equal to

8

and the length of its conjugate axis is equal to half of the distance between its foci, is :

A
${2 \over {\sqrt 3 }}$
B
${\sqrt 3 }$
C
${{4 \over 3}}$
D
${4 \over {\sqrt 3 }}$

View written solutionFree

Correct answer: A

Write the standard form and basic relations

For a hyperbola in standard form $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,$ we use:

Eccentricity: $e=\frac{c}{a}, \quad c^2=a^2+b^2$
Length of conjugate axis: $2b$
Distance between foci: $2c$
Length of latus rectum: $\frac{2b^2}{a}$

Use the condition involving conjugate axis and foci

Given:

length of conjugate axis is equal to half of the distance between its foci.

So, $2b=\frac{1}{2}(2c)=c$ Hence, $c=2b$

Now use $c^2=a^2+b^2$ Substituting $c=2b$ : $(2b)^2=a^2+b^2$ $4b^2=a^2+b^2$ $a^2=3b^2$

Therefore, $\frac{b^2}{a^2}=\frac{1}{3}$

Use the latus rectum condition

Given latus rectum length is $8$ : $\frac{2b^2}{a}=8$ $b^2=4a$

This relation is not actually needed to find $e$ , because we already have the ratio between $a$ and $b$ from Step 2.

Find the eccentricity

Since $e^2=\frac{c^2}{a^2}=\frac{a^2+b^2}{a^2}=1+\frac{b^2}{a^2},$ we get $e^2=1+\frac{1}{3}=\frac{4}{3}$ Thus, $e=\frac{2}{\sqrt{3}}$

Match with the options

$\frac{2}{\sqrt{3}}$ corresponds to Option A.

Compare with stored correct answer

Stored correct answer: A

Our derived answer: A

So the answer agrees with the stored correct answer.

More from Hyperbola

For the Hyperbola $\frac{x ^{2}}{c o s ^{2} α} - \frac{y ^{2}}{s i n ^{2} α} = 1$ , which of the following remains constant when $α$ varies $=$ ?2007 · MCQ
The foci of the ellipse $\frac{x ^{2}}{16} + \frac{y ^{2}}{b ^{2}} = 1$ and the hyperbola $\frac{x ^{2}}{144} - \frac{y ^{2}}{81} = \frac{1}{25}$ coincide. Then the value of $b^{2}$ is :2003 · MCQ
Let one focus of the hyperbola $H : \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ be at $(10, 0)$ and the corresponding directrix be $x = \frac{9}{10}$ . If $e$ and $l$ respectively are the eccentricity and the…2025 · MCQ
Let the product of the focal distances of the point $P (4, 23)$ on the hyperbola $H : \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ be 32 . Let the length of the conjugate axis of H be $p$ and the length of its latus rectum be $q$ …2025 · Numerical
If the equation of the hyperbola with foci $(4, 2)$ and $(8, 2)$ is $3 x^{2} - y^{2} - α x + β y + γ = 0$ , then $α + β + γ$ is equal to .2025 · Numerical
Let the sum of the focal distances of the point $P (4, 3)$ on the hyperbola $H : \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ be $8 \frac{5}{3}$ . If for H , the length of the latus rectum is $l$ and the…2025 · MCQ
Consider the hyperbola $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ having one of its focus at $P (- 3, 0)$ . If the latus ractum through its other focus subtends a right angle at P and $a^{2} b^{2} = α 2 - β, α, β \in N$ …2025 · Numerical
Let e1 and e2 be the eccentricities of the ellipse $\frac{x ^{2}}{b ^{2}} + \frac{y ^{2}}{25} = 1$ and the hyperbola $\frac{x ^{2}}{16} - \frac{y ^{2}}{b ^{2}} = 1$ , respectively. If b < 5 and e1e2 = 1, then the eccentricity of the ellipse having its…2025 · MCQ