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Question

Find the locus of the point of intersection of two normals to a parabola which are at right angles to one another.

Solution
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The equation of the normal to the parabola y2=4ax is y=mx2amam3.

It passes through the point (h,k) is

k=mh2amam3

am3+m(2ah)+k=0 .....(1)

Let the roots of the above equation be m1,m2,m3. Let the perpendicular normals correspond to the values of m1 and m2 so that m1m2=1.

From equation 1, m1m2m3=ka

Since, m1m2=1,m3=ka

Since m3 is a root of equation 1, we have

a(ka)3+ka(2ah)+k=0

k2+a(2ah)+a2=0

k2=a(h3a)

Hence, the locus of (h,k) is y2=a(x3a).

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