The locus of the midpoints of the portion of the normal to the parabola `y^(2)=16x` intercepted between the curve and the axis is another parabola whose latus rectum is ___________ .

Answer

4

Explanation

(4)
Consider the parabola `y^(2)=4ax`.
We have to find the locus of R(h,k). Since Q ordinate 0, the ordinate of P is 2k.
Also, P is on the curve. Then the abscissa of P is `k^(2)//a`.
Now, PQ is normal to the curve.
slope of tangent to the curve at any point is
`(dy)/(dx)=(2a)/(y)`
Hence, the slope of normal at point P is `-k//a`.
Also, the slope of normal joining P and R(h,k) is
`(2k-k)/((k^(2)//a)-h)`
Hence, comparing slopes, we get
`(2k-k)/((k^(2)//a)-h)=-(k)/(a)`
`ory^(2)=a(x-a)`
For `y^(2)=16x,a=4`. Hence, the locus is `y^(2)=4(x-4)`.