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For an ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ ($a > b$), the **length of the latus rectum** is:
A$\dfrac{2a^2}{b}$
B$\dfrac{2b^2}{a}$
C$\dfrac{a}{b}$
D$2a$
Answer & Solution
Correct answer: B. $\dfrac{2b^2}{a}$
**Latus rectum** is the chord through a focus perpendicular to the major axis.
For the horizontal ellipse, plug $x = c$ into the standard equation:
$\dfrac{c^2}{a^2} + \dfrac{y^2}{b^2} = 1 \Rightarrow \dfrac{y^2}{b^2} = 1 - \dfrac{c^2}{a^2} = \dfrac{a^2 - c^2}{a^2} = \dfrac{b^2}{a^2}$.
So $y^2 = \dfrac{b^4}{a^2}$, $y = \pm \dfrac{b^2}{a}$. Total chord length = $\dfrac{2b^2}{a}$.
**Trap.** Option A swaps $a$ and $b$. Option D ($2a$) is the major-axis length, not the latus rectum.
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