Intersection (Geometric) Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Intersection (Geometric).

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The set of all points where two or more geometric objects (lines, planes, curves) meet or cross each other.

Where two roads cross—that single crossing point is their intersection.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: An intersection is the set of points two or more figures share — solved by satisfying all their equations at once.

Common stuck point: The procedure for intersection (geometric) is the easy part; the trap is assuming a unique crossing point. Asking "Am I looking for the point(s) that lie on two or more figures at the same time?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I looking for the point(s) that lie on two or more figures at the same time?

Worked Examples

Example 1

easy

Find the intersection point of lines

\ell_1: y = 3x - 2

and

\ell_2: y = -x + 6

Find the intersection point of the two lines.

Answer

Intersection at

(2, 4)

First step

Step 1: Set equations equal (both equal

y

3x - 2 = -x + 6

Full solution

2
Step 2: Solve: $4x = 8 \Rightarrow x = 2$ .
3
Step 3: Substitute back: $y = 3(2) - 2 = 4$ . Check: $y = -(2) + 6 = 4$ . ✓

Two non-parallel lines in a plane intersect at exactly one point, found by solving their equations simultaneously. Substituting back into both equations verifies the solution is correct.

Example 2

medium

Find the intersection point(s) of line

y = x + 3

and circle

x^2 + y^2 = 25

. Classify the intersection (secant, tangent, or no intersection).

Example 3

medium

Find the intersection points of

y = x

and the circle

x^2 + y^2 = 18

Example 4

hard

Find the points where

y = x^2

and

y = 6 - x

intersect.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Do lines

y = 2x + 3

and

y = 2x - 5

intersect? Explain why or why not.

Example 2

hard

Find the intersection of the two circles:

C_1: x^2 + y^2 = 25

and

C_2: (x-4)^2 + y^2 = 9

Example 3

easy

Two non-parallel lines in a plane intersect in how many points?

Example 4

easy

Find the intersection of

y = x

and

y = -x + 2

Find the intersection of y=x and y=−x+2.

Example 5

easy

Do the parallel lines

y = 2x + 1

and

y = 2x + 5

have an intersection?

Example 6

easy

Two distinct planes in 3D intersect in what?

Example 7

easy

A line can cross a circle in at most how many points?

Example 8

easy

Where does the line

y = 3x - 6

cross the

x

-axis?

Where does y=3x−6 cross the x-axis?

Example 9

easy

Two circles can intersect in at most how many points?

Example 10

easy

Where do the lines

x = 4

and

y = 7

intersect?

Find where x=4 and y=7 intersect.

Example 11

medium

Find the intersection of

y = 2x + 1

and

y = -x + 7

Find the intersection of the two lines.

Example 12

medium

How many intersection points do 4 lines have if no two are parallel and no three meet at a point?

Example 13

medium

Find where the line

y = 0

meets the circle

x^2 + y^2 = 25

Find where y=0 meets the circle x²+y²=25.

Example 14

medium

A system of two line equations gives '

0 = 5

' when solved. What does this mean geometrically?

Example 15

medium

Find the intersection of

2x + y = 7

and

x - y = 2

Example 16

medium

When does a line touch a circle at exactly one point?

Example 17

medium

A system of two line equations has infinitely many solutions. What does this mean geometrically?

Example 18

medium

Three lines pass through a single common point. How many distinct intersection points are there?

Example 19

challenge

Find the intersection points of the line

y = x + 1

and the circle

x^2 + y^2 = 25

Example 20

challenge

For what value of

c

is the line

y = x + c

tangent to the circle

x^2 + y^2 = 8

Example 21

challenge

Two circles,

x^2 + y^2 = 4

and

(x-3)^2 + y^2 = 4

, intersect. Find the

x

-coordinate of their intersection points.

Example 22

challenge

Explain why the intersection of two surfaces in 3D is generally a curve, not a point, using dimensions.

Example 23

easy

Find the intersection point of

y = x + 3

and

y = 2x + 1

Find the intersection of the two lines.

Example 24

easy

Where does

y = 4x - 8

cross the

x

-axis?

Example 25

easy

Find where the lines

x = 5

and

y = -3

cross.

Find where x=5 and y=−3 cross.

Example 26

easy

Find the intersection of

y = -2x + 6

and the

y

-axis.

Example 27

medium

Find the intersection of

3x + 2y = 12

and

x - y = 1

Example 28

medium

Find the intersection of

y = x^2

and

y = 4

Example 29

medium

How many intersection points do 5 lines have if no two are parallel and no three meet at a point?

Example 30

medium

For what value of

k

do the lines

y = 2x + 3

and

y = kx - 1

NOT intersect?

Example 31

medium

Find the intersection of

y = x^2 - 1

and

y = 2x + 2

Example 32

medium

Find the intersection of

x + y = 4

and

x^2 + y^2 = 16

Example 33

medium

Find the intersection of

y = 2x - 1

and

y = -x + 5

algebraically.

Find the intersection of the two lines algebraically.

Example 34

hard

Find the intersection points of the circles

x^2 + y^2 = 25

and

(x-7)^2 + y^2 = 18

Example 35

hard

For what value(s) of

c

is the line

y = 2x + c

tangent to

x^2 + y^2 = 5

Example 36

hard

Where does

y = x^3

meet

y = x

Example 37

hard

Lines

y = mx

and

y = -x + 6

meet on the line

x = 2

. Find

m

Example 38

hard

Find all intersection points of

y = \sin x

and

y = \tfrac{1}{2}

[0, 2\pi]

Example 39

hard

Two segments

[A,B]

and

[C,D]

lie on lines that cross at

P

. Does

P

always lie on both segments?

Example 40

challenge

For what

k

does the line

y = kx + 2

touch the parabola

y = x^2

at exactly one point?

Example 41

challenge

How many intersection points can

n

distinct circles have at most in a plane, assuming each pair intersects in two points?

Example 42

challenge

Two circles have centers

(0,0)

and

(d,0)

, both with radius

r

. For which

d

do they intersect in exactly one point?

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Related Concepts

Line Systems of Equations

Background Knowledge

These ideas may be useful before you work through the harder examples.

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