Parabola

If the parabolas y²=4a(x-b) and x²=4a(y-c) touch each other at right angles, the locus of their point of contact in terms of "a" is what?

4 Answers

Touch Each other should probably be "Intersect each other"

y²=4a(x-b) and x²=4a(y-c)

dy/dx=2a/y also dy/dx = x/2a

at the common point of intersection, the slopes are
2a/y. x/2a = -1
x=-y

now substitute in the original equation

x²=4a(x-b) and x²=4a(x-c)

so first we get that b=c is necessary ...

also
x²-4ax+4ab = 0
x={4aÂ±√16a²-16ab}/2
x=2aÂ±2√a²-ab

now Please check what the touch each other at 90 degrees means!!

is there any alternative method sir?

Nikhil,

I cant think of any other method.

The options were:
a. xy=4a²
b. xy=4a
c. xy=a
d. None

Answer is A.

Could you please explain it sir?

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