Answer to Question #157485 in Calculus for Keleko

Question

Given that the equation of tangent at the point (at^2 , 2at) on the parabola y^2 = 4ax is x - ty + at^2 = 0
A(at1^2 , 2at1) and B(at2^2 , 2at2) are points on this parabola. The equation of the chord AB is 2x - (t1 + t2)y + 2at1at2 = 0
The tangents A and B meet at the point P. Find the coordinates of P.
The line through P parallel to the axis of the parabola meets the chord AB at M. Find the coordinates of M.

Answer 1

The tangent A

$$x - t_1y + at_1^2 = 0$$

The tangent B

$$x - t_2y + at_2^2 = 0$$

The tangents A and B meet at the point P

$$t_1y - at_1^2 =t_2y - at_2^2$$

Since $t_1\not=t_2$

$$y=\dfrac{a(t_2^2-t_1^2)}{t_2-t_1}$$

$$y=a(t_2+t_1)$$

$$x=\dfrac{y^2}{4a}=\dfrac{a(t_1+t_2)^2}{4}$$

$P\big(\dfrac{a(t_1+t_2)^2}{4}, a(t_2+t_1)\big)$

The equation of the axis of the parabola: $y=0.$

The equation of the line through P parallel to the axis of the parabola:

$$y=a(t_1+t_2)$$

The equation of the chord AB is

$$2x-(t_1+t_2)y+2at_1t_2=0$$

The line through P parallel to the axis of the parabola meets the chord AB at M

$$2x-(t_1+t_2)(a(t_1+t_2))+2at_1t_2=0$$

$$2x=a(t_1^2+2at_1t_2+t_2^2-2t_1t_2)$$

$$x=\dfrac{1}{2}(at_1^2+at_2^2))$$

$M\big(\dfrac{1}{2}(at_1^2+at_2^2)), \dfrac{1}{2}(2at_1+2at_2))\big)$

Therefore M is the midpoint of AB.