Parametric representation - locus with one point (1 Viewer)

[KK10047]

The tangent to the parabola x^2 = 4ay at P(2ap, ap^2) cuts the x-axis at A and the y-axis at B. Find the coordinates of A and B. Find the equation of the locus of M, the mid-point of AB.

Answer is y = -2x^2/a

 

[pikachu975]

y' = x/2a
= p
Tangent equation:
y - ap^2 = p(x-2ap)
y = px - ap^2

A occurs when y=0
p(x-ap) = 0
x = ap

B occurs when x = 0
y = -ap^2

So A(ap, 0) and B(0, -ap^2)

M = (ap/2 , -ap^2 /2)
So x = ap/2 and y = -ap^2 /2
x^2 = a^2p^2 / 4
Divide y by x^2
y/x^2 = (-ap^2 /2) * (4/a^2 p^2)
y/x^2 = -2/a
y = -2x^2 / a