Geometry using polynomial techniques

Use sums and products of roots to find midpoints of chords, chord lengths and conditions for tangency when a line meets a parabola.

The method

When a line meets a curve, the $x$-coordinates of the intersection points are the roots of the equation you get by solving the two simultaneously. You often do not need those roots at all, only their sum and product.

Chords of y = x²

The midpoint M of the chord always has x = m/2, half the sum of the roots. Keep m fixed and change c: M slides up a vertical line.

Try this

Find a value of $c$ that makes the line a tangent when $m = 2$. What does the readout say?

Midpoints of chords

Example 1●●

The line $y = 2x + 3$ meets $y = x^2$ at $A$ and $B$. Find the midpoint of $AB$.

$x^2 - 2x - 3 = 0$, so $\alpha + \beta = 2$.

$x = \dfrac{\alpha + \beta}{2} = 1$

The midpoint's x-coordinate is the average of the roots.

$y = 2(1) + 3 = 5$, so the midpoint is $(1, 5)$.

The midpoint lies on the line.

Chord lengths

Example 2●●●

Find the length of the chord in Example 1.

$(\alpha - \beta)^2 = 2^2 - 4(-3) = 16$, so $|\alpha - \beta| = 4$.

Length $= \sqrt{1 + 2^2} \times 4 = 4\sqrt5$.

Tangents

Example 3●●

Find $c$ so that $y = 4x + c$ is a tangent to $y = x^2$, and find the point of contact.

$x^2 - 4x - c = 0$ needs $\Delta = 16 + 4c = 0$, so $c = -4$.

$x = 2$, so the point of contact is $(2, 4)$.

A double root α = β, and α + β = 4.

Flashcards

Midpoint $x$ of a chord

$\dfrac{\alpha + \beta}{2}$

$(lpha - eta)^2$

$(\alpha + \beta)^2 - 4\alpha\beta$

Tangent condition

The intersection equation has a double root: $\Delta = 0$.

Practice

The line $y = 2x + 3$ meets $y = x^2$ at $A$ and $B$. Find the $x$-coordinate of the midpoint of $AB$.
Find the $y$-coordinate of that midpoint.
For $x^2 - 2x - 3 = 0$, find $(\alpha - \beta)^2$.
Find $c$ so that $y = 4x + c$ is a tangent to $y = x^2$.
Find the $x$-coordinate of the point where $y = 4x - 4$ touches $y = x^2$.
A line is tangent to a parabola exactly when the intersection equation has:
The line $y = 6x - 1$ meets $y = x^2$ where $x = \alpha$ and $x = \beta$. Find $\alpha + \beta$.
The midpoints of all chords of $y = x^2$ with gradient $m$ lie on:
The chord from $x = \alpha$ to $x = \beta$ on the line $y = mx + c$ has length:
$y = mx + c$ meets $y = x^2$ at two distinct points exactly when: