Geometry of parabola Chord of Contact: Let T(2at,at^2) be any point on parab. x^2=2ay

[Jeane]

I am working through my son's Year 11 text on my own and am working on the section on the above. I have a specific query but will list the whole question
so you know what it is all about.

Let T(2at , at²) be any point on the parabola x² = 2ay
(a) Show that the tangent at T has equation y = tx - at²

This I can do.

(b) If this tangent passes through the point P(x₀, y₀), show that at² - x₀t + y₀ = 0

This I can do

(c) What is the condition for this quadratic equation in t to have two real roots?

x₀² > 2ay

(d) Suppose that t ₁ and t ₂ are the roots of the quadratic equation and let T₁ and T₂ be the points on the parabola corresponding to t = t₁ and t = t₂, respectively.
(i) Show that the Chord T₁T₂ has equation (t₁ + t₂) x = 2y + 2at₁t₂.

At first sight this seems easy. First find m by using (y - y₁)/(x - x₁) which is (t₁ + t₂).
so to find the terms to fill the formula y = mx + b, I used (y - y₁) = m(x - x₁) with X₁ and Y₁ being (2at₁, at₁²)
I got the equation

(t₁ + t₂)x = y + at₁² + 2at₁t₂

...which is close and would work if y = at₁² which it does
but can it be equated with the unkown y logically? Is there another way to come at this?

(ii) Show that t₁ + t₂ = x₀/a and t₁t₂ = y₀/a

This one I don't know how to solve

Hence (iii) show that the chord T₁T₂ has equation x₀x = 2a(y + y₀)

I assume this would follow logically from the last step