Answer to Question #157488 in Calculus for Ivo

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. Prove that M is the midpoint of AB.

Answer 1

$x=\frac{y^2}{4a},$

$k=x_y'=\frac{y}{2a},$

$x-at_i^2=\frac{2at_i}{2a}(y-2at_i),$

$\begin{cases} x-at_1^2=t_1(y-2at_1), \\ x-at_2^2=t_2(y-2at_2); \end{cases}$

$\begin{cases} x_0=at_1t_2, \\ y_0=a(t_1+t_2); \end{cases}$ – coordinates of point P.

Line PM (parallel to the x-axis) has an equation $y=a(t_1+t_2).$

Midpoint of AB has coordinates $(\frac a2(t_1^2+t_2^2),~a(t_1+t_2)).$

AB equation: $2x-(t_1+t_2)y+2at_1t_2=0,$

with a PM equation we'll have

$\begin{cases} x_0=\frac a2(t_1^2+t_2^2), \\ y_0=a(t_1+t_2); \end{cases}$ – coordinates of point M, and they are equal to the coordinates of point P.