Question

For the ellipse \[3x^2+4y^2=12,\] the length of latus rectum is______.

पर्याय

• \[\frac{3}{2}\]

• 3

• \[\frac{8}{3}\]

• \[\sqrt{\frac{3}{2}}\]

Answer 1

For the ellipse \[3x^2+4y^2=12,\] the length of latus rectum is 3.

Explanation:

The given ellipse is: \[3x^2+4y^2=12,\]

Dividing both sides by 12:

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

Here:

• \[a^2=4\Rightarrow a=2\] (major axis along x-axis)
• \[b^2=3\Rightarrow b=\sqrt{3}\]

The formula for length of latus rectum is:

                                                                \[L=\frac{2b^2}{a}\]

                                                               \[L=\frac{2\times3}{2}=\frac{6}{2}=3\]