Circles

Line Circle Intersection

Find the intersection points of a line and circle by substitution and verify both equations.

line-circle-intersection Substitute the line equation into the circle equation, solve the resulting quadratic, then verify each point.

Example

Solve the intersection points of a line and circle.

highlighted = computed this step

Step 1 — Plot circle and line

Plot the line and circle intersections.

Coordinate plot\text{Coordinate plot}
Line-circle intersection plotLine y=3 intersects the circle at A(-4, 3) and B(4, 3).lineAB

Step 2 — Circle equation

Read the circle equation.

x2+y2=25x^{2} + y^{2} = \hl{25}

Step 3 — Substitute the line

Read the horizontal line.

y=3y= \hl{3}

Step 4 — Substitute y

Substitute the line into the circle.

x2+32=25x^{2} + \hl{3} ^{2} = 25

Step 5 — Solve for x squared

Solve for x squared.

x2=16x^{2} = \hl{16}

Step 6 — Roots

Take both square roots.

x=-4,4x= \hl{-4} , \hl{4}

Step 7 — Intersection points

State the intersection points.

2 points: A=(-4,3)B=(4,3)\hl{2} \text{ points: } A=( \hl{-4} , \hl{3} ) \quad B=( \hl{4} , \hl{3} )