CircleMedium
Question
Two parabolas y2 = 4a(x -i1) and x2 = 4a(y i2) always touch one another, the quantities i1 and i2 are both variable. Locus of their point of contact has the equation -
Options
A.xy = a2
B.xy = 2a2
C.xy = 4a2
D.none
Solution
Let P(x1 y1) be point of contact of two parabola.
Tangents at P of the two parabolas are
yy1 = 2a(x + x1) - 4al1 and
xx1 = 2a(y + y1) - 4al2
⇒ 2ax - yy1 = 2a (2l1 - x1) ... (i)
and xx1 - 2ay = 2a (y1 - 2l2) ... (ii)
clearly (i) and (ii) represent same line
∴
⇒ x1y1 = 4a2
Hence locus of P is xy = 4a2
Tangents at P of the two parabolas are
yy1 = 2a(x + x1) - 4al1 and
xx1 = 2a(y + y1) - 4al2
⇒ 2ax - yy1 = 2a (2l1 - x1) ... (i)
and xx1 - 2ay = 2a (y1 - 2l2) ... (ii)
clearly (i) and (ii) represent same line
∴
⇒ x1y1 = 4a2Hence locus of P is xy = 4a2
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