Question

Three normals from a point to the parabola y² = 4ax meet the axis of the parabola in points whose abscissa are in A.P. Find the locus of the point.

Answer 1

The equation of any normal to the parabola is

y = mx – 2am – am³

It passes through the point (h, k) if

am³ + m (2a - h) + k = 0 ... (1)

The normal cuts the axis of the parabola viz., y = 0 at point where x = 2a + am²

Hence the abscissa of the points in which the normal  through (h, k) meet the axis of the parabola are

x₁ = 2a + am₁ ² , x₂ = 2a + am₂ ² , x₃ = 2a + am₃ ² 

Since x₁ , x₂ , x₃ are in A.P.

(2a + am²₁ ) + (2a + am₃² ) = 2 (2a + am₂² )

m²₁ + m²₂ = 2m²₂ ... (2)

Also, from (1), m₁ + m₂ + m₃ = 0.

m₂m3 + m₃m₁ + m₁m₂  = 2a - h/a

and m₁ m₂ m₃ = – k/a ... (5)

From (3),

Hence the locus of (h, k) is 27 ay² = 2(x – 2a)³ .