Systems with a line and a conic

Warm-up

Three quick questions from earlier lessons.

Discover

How many points can a line and a circle have in common?

0, 1 or 2
Always 2

Key idea

To find where a line meets a conic, solve the two equations together.

Substitute the line into the conic: you get a quadratic in one variable. Its real solutions give the intersection points: 2 solutions, 2 points; a double root, a tangent; no real root, no intersection. Then find the other coordinate from the line.

Worked example

Where does $y = x + 1$ meet $x^2 + y^2 = 25$ ?

x^2 + (x + 1)^2 = 25

Substitute y.

2x^2 + 2x - 24 = 0

, so

x^2 + x - 12 = 0

Expand and simplify.

(3, 4)

and

(-4, -3)

(x + 4)(x − 3) = 0, then y = x + 1.

Your turn, guided

Your turn: where does $y = 3$ meet $x^2 + y^2 = 25$ ?

x^2 = 16

Substitute y = 3.

x = 4

or

x = -4

Two square roots: type the negative one.

Practice ★

The circle $x^2 + y^2 = 169$ and the line $y = 12$ meet at two points. Give their $x$ -coordinates.
The circle $x^2 + y^2 = 25$ and the line $y = 4$ meet at two points. Give their $x$ -coordinates.
The circle $x^2 + y^2 = 400$ and the line $y = 16$ meet at two points. Give their $x$ -coordinates.

Watch out!

Practice ★★

How many points do the circle $x^2 + y^2 = 16$ and the line $y = 11$ have in common?
1. None
2. One
3. Two

Find the $x$ -coordinates of the points where $y = x^2$ meets $y = x + 6$ .
Find the $x$ -coordinates of the points where $y = x^2$ meets $y = x + 20$ .
Find the $x$ -coordinates of the points where $y = x^2$ meets $y = x + 2$ .

Exit ticket

Three questions to check you have got it.

Find the $x$ -coordinates of the points where $y = x^2$ meets $y = x + 6$ .
Find the $x$ -coordinates of the points where $y = x^2$ meets $y = x + 20$ .
Find the $x$ -coordinates of the points where $y = x^2$ meets $y = x + 2$ .

Summary card

A line and a conic

Substitute the line into the conic.
Solve the quadratic: 0, 1 or 2 real solutions.
Find the other coordinate from the line.

Watch out: a negative square means no intersection.

About this lesson