Show that the locus of the midpoints of chords of a parabola passing through the vertex is in turn a parabola whose latus rectum is half of the latus rectum of the original.
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Show that the locus of the midpoints of chords of a parabola passing through the vertex is in turn a parabola whose latus rectum is half of the latus rectum of the original.
Show that the locus of the midpoints of chords of a parabola passing through the vertex is in turn a parabola whose latus rectum is half of the latus rectum of the original.
The equation of the chord in terms of its midpoint M(x1, y1) is
S1 = S11
⇒ yy1 – 2a(x + x1) = y12 – 4ax1
yy1 – 2ax = y12 – 2ax1 which passes through (0,0)
y12 – 2ax1 = 0
Hence, the locus of M(x1, y1) is the parabola y2 = 2ax.