Question

The length of the latus rectum of the hyperbola \[3x^{2}-y^{2}=4\] is______.

Options

• \[8\sqrt{3}\]

• \[4\sqrt{3}\]

• 16

• 32

Answer 1

The length of the latus rectum of the hyperbola \[3x^{2}-y^{2}=4\] is \[4\sqrt{3}\].

Explanation:

Rewrite in standard form:

                     \[3x^2-y^2=4\Rightarrow\frac{x^2}{\frac{4}{3}}-\frac{y^2}{4}=1\]

So \[a^2=\frac{4}{3},b^2=4.\]

Length of latus rectum of hyperbola \[=\frac{2b^2}{a}{:}\]

                    \[a=\sqrt{\frac{4}{3}}=\frac{2}{\sqrt3}\]

Latus rectum = \[\frac{2\times4}{\frac{2}{\sqrt{3}}}=\frac{8}{\frac{2}{\sqrt{3}}}=8\times\frac{\sqrt{3}}{2}=4\sqrt{3}\]