Question
If from the vertex of a parabola $y^{2} = 4ax$ a pair of chords be drawn at right angles to one another and with these chords as adjacent sides a rectangle be made, then the locus of the further angle of the rectangle is :
Options
Solution
Sol. [Key Concept: With two perpendicular vertex-chords as adjacent sides of a rectangle, the far corner traces a parabola; eliminate the two parameters to find its focus/directrix.]
Let the parameters of the two chords be t and s. Since the chords are perpendicular,
(2/t)(2/s) = −1 ⇒ ts = −4.
If P(x, y) is the fourth vertex of the rectangle, then
x = a(t² + s²), y = 2a(t + s).
Since ts = −4,
t² + s² = (t + s)² + 8.
Therefore, x = a(t + s)² + 8a and y = 2a(t + s).
Eliminating (t + s),
y² = 4a(x − 8a).
This is a parabola of the same focal parameter as y² = 4ax, hence it is an equal parabola.
Its vertex is (8a, 0) and its focal length is a.
Therefore, its focus is (9a, 0) and its directrix is x = 7a.
Hence, the correct options are: (a), (c).
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