Question

Find the eccentricity of the conic $3x^2+4y^2=24$.

• A. $\frac{1}{4}$
• B. $\frac{7}{4}$
• C. $\frac{1}{2}$
• D. $\sqrt{\frac{7}{4}}$

Answer 1

The correct option is C $\frac{1}{2}$

The equation of conic,

$\Rightarrow$ $3x^2+4y^2=24$

$\Rightarrow$ $\frac{3x^2}{24}+\frac{4y^2}{24}=1$ [ Dividing both sides by $24$ ]

$\Rightarrow$ $\frac{x^2}{8}+\frac{y^2}{6}=1$

Comparing above equation with $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$ we get:

$\Rightarrow$ $a^2=8$ and $b^2=6$

We know, $b^2=a^2(1-e^2)$

$\Rightarrow$ $6=8(1-e^2)$

$\Rightarrow$ $1-e^2=\frac{6}{8}$

$\Rightarrow$ $1-e^2=\frac{2}{4}$

$\Rightarrow$ $e^2=1-\frac{3}{4}$

$\Rightarrow$ $e^2=\frac{4-3}{4}$

$\Rightarrow$ $e^2=\frac{1}{4}$

$\therefore$ $e=\frac{1}{2}$

$\therefore$ The required eccentricity is $\frac{1}{2}.$