Question

Smallest focal chord of a parabola

Answer 1

Latus Rectum

Answer 2

A focal chord of a parabola is any chord that passes through the focus. The latus rectum is a special focal chord that is perpendicular to the axis of symmetry. For a parabola $y^2 = 4ax$, the length of a focal chord making an angle $\theta$ with the axis is $L = 4a \csc^2 \theta$. The minimum length occurs when $\csc^2 \theta$ is minimized, which is 1 when $\theta = \frac{\pi}{2}$. This corresponds to the latus rectum, whose length is $4a$.