Parametrics Problems! (1 Viewer)

[Finx]

Does anyone else find parametrics a hard topic to grasp? If not, is there some easy way to go about it, because its giving me (and a majority of my class) quite a hassle.

Here's a question.
Tangents from the point P(x1, y1) touch the parabola x^2 = 8y at the points A and B.
a) Show that the x-coordinates of A and B are the roots of the quadraticc equation x^2 - 2x1x + 8y1.

The 1 in 2x1x is subscript, as is the 1 in 8y1.

Thanks in advance!

 

[Trebla]

x² = 8y
(Note that a = 2)
dy/dx = x/4
Let A be (4p, 2p²) and B be (4q, 2q²).
Equation of tangent at A:
(1) y - 2p² = p(x - 4p)
Equation of tangent at B:
(2) y - 2q² = q(x - 4q)

(1) - (2) = 2(q² - p²) = x(p - q) + 4(q² - p²)
x(p - q) = 2(p² - q²)
x = 2(p + q)
sub into (1)
y - 2p² = p(2p + 2q - 4p)
y - 2p² = - 2p² + 2pq
y = 2pq
Since the intersection of tangents occurs at (x₁, y₁), we have
x₁ = 2(p + q)
=> 2x₁ = 4p + 4q
Also,
y₁ = 2pq
=> 8y₁ = 16pq = (4p)(4q)
The x-coordinates of A and B are 4p and 4q respectively and we have
4p + 4q = 2x₁
(4p)(4q) = 8y₁
These are the sum of roots and products of roots of the quadratic equation:
x² - 2xx₁ + 8y₁ = 0