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Find the locus of the midpoints of the p...

Find the locus of the midpoints of the portion of the normal to the parabola `y^2=4a x` intercepted between the curve and the axis.

Text Solution

Verified by Experts

The correct Answer is:
`y^(2)=a(x-a)`
Normal at `P(at^(2),2at)" is "t=-tx+2at+at^(3)`.
It meets the axis y=0 at `G(2a+at^(2),0)`.
If (x,y) is the midpoint of PG, then
`2x=2a+at^(2)+at^(2),2y=2at`
Eliminating t, we have
`x-a=at^(2)-a((y)/(a))^(2)`
`or" "y^(2)=a(x-a)`
which is the required locus.
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