Mathematics

Problem

Let

P(x_{1},y_{1})

be any point on the parabola

y^{2}=4ax

with focus

F(a,0)

. The point

P'(x_{1},-y_{1})

is the reflection of

P

in the axis of the parabola. a. (i) Prove that the distances

PF

and

P'F

are equal. b. (ii) Show that the segment

PP'

is perpendicular to the axis and that its midpoint lies on the axis. c. (iii) Using your results, explain why every pair of reflected points on the parabola are equidistant from the focus.

Intuitive explanation

The key insight is recognizing that the parabola's defining property—equal distance from focus and directrix—applies to both $P$ and its reflection $P'$ symmetrically. Since $P'$ is simply $P$ flipped across the $x$ -axis, it also lies on the parabola, and the distances $|PF|$ and $|P'F|$ are equal by symmetry. This reflection property is fundamental to understanding how the parabola's geometry behaves under this transformation.

Intuitive explanation

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