Two parabolas are given by and with

MATHEMATICS · Subjective · AP · AP Precalculus · Functions with Parameters, Vectors, and Matrices · Functions with Parameters, Vectors, and Matrices

Problem

Two parabolas are given by

y^{2}=4ax

and

y^{2}=-4ax

with

a>0

. (i) On the same set of coordinate axes, sketch both parabolas and label their vertices, axes, and foci. (ii) For an arbitrary ordinate value

y_{0}

, the two parabolas give points

A\bigl(x_{1},y_{0}\bigr)

and

B\bigl(x_{2},y_{0}\bigr)

. Prove that the midpoint of

AB

always lies on the y-axis. (iii) If

a=3

and

y_{0}=6

, find the coordinates of

A

B

, and the midpoint obtained in part (ii), and verify your result numerically.