An ellipse has its center at (1, -2), one focus at (3, -2) and one vertex at (5, -2).

Question

An ellipse has its center at $(1, -2)$, one focus at $(3, -2)$ and one vertex at $(5, -2)$. Then the length of its latus rectum is :

Accepted Answer

Given

Center of ellipse : $(1,-2)$

Focus : $(3,-2)$

Vertex : $(5,-2)$

The center, focus, and vertex all have the same $y$-coordinate $(-2)$, which means the major axis of the ellipse is horizontal (parallel to the $x$-axis).

Find $a$ (semi-major axis)

Distance from center to vertex :

$a=|5-1|=4$

Find $c$ (distance from center to focus)

$c=|3-1|=2$

Find $b$ using ellipse relation

${c}^{2}={a}^{2}-{b}^{2}$

$\Rightarrow$ ${2}^{2}={4}^{2}-{b}^{2}$

$\Rightarrow 4=16-{b}^{2}$

$\Rightarrow {b}^{2}=12$

$\Rightarrow b=2\sqrt{3}$

Length of latus rectum

For an ellipse :

Length of latus rectum $=\frac{2{b}^{2}}{a}$

$=\frac{2(12)}{4}=6$

Solutions · 1