Prove geometrically that subnormal to the parabola y^2=4ax is a constant?

Answer 1

y²=4ax

Parametric form is
x = at²
y = 2at

Equation of normal at (at², 2at) is given by

y + tx = 2at + at³

Let the subnormal at (at², 2at) be PN

P is the x coordinate of the point P
Thus P = (at², 0)

N is found where the normal cuts the x-axis ie when y = 0
Thus at N:

0 + tx = 2at + at³
ie x = 2a + at²

ie N = (2a + at², 0)

Thus length of PN = 2a + at² - at²
= 2a, which is independent of N ie it is constant (AND equals half the length of the latus rectum)

Answer 2

Proof here, about 3 pages or so down. Look for "about parabola".

http://www.ping.be/~ping1339/Pana.htm

The constant, if I read it right, for your parabola should be 2a.