Line and Curve Intersection Finder

Math Geometry • Coordinate Geometry

Written by STEM Calculators Team Published January 23, 2026 Updated February 24, 2026

Find intersection points between lines and common curves (circle, parabola), with steps and a pan/zoom graph.

Inputs accept 1e-3, pi, e, sqrt(2), sin(), cos(), tan(), ln(), log(), abs(). Use * for multiplication.

Select intersection type

Intersection:

Tip: Line–circle usually yields 0, 1 (tangent), or 2 intersection points.

Line input

Line form:

Slope \(m\):

Intercept \(b\):

\(A\):

\(B\):

\(C\):

Circle input

Circle form:

Center \(h\):

Center \(k\):

Radius \(r\):

\(D\):

\(E\):

\(F\):

If you use center-radius, the tool converts to the general form to solve robustly.

Second line input (for Two Lines)

Line 2 form:

Slope \(m_2\):

Intercept \(b_2\):

\(A_2\):

\(B_2\):

\(C_2\):

Parabola input (for Line & Parabola)

Parabola is in the upright form \(y=ax^2+bx+c\) (not rotated).

\(a\):

\(b\):

\(c\):

Graph options

Axis units label: Show grid: Show tick labels:

The plot uses square units (equal scale in x and y), so intersections look geometrically correct.

Ready

Choose an intersection type and click Calculate.

Intersection diagram (pan/zoom enabled)

Drag to pan • wheel/trackpad to zoom • pinch on touch • “Reset view” fits the geometry.

Rate this calculator

0.0 /5 (0 ratings)

Be the first to rate.

Your rating

Name (optional) Review (optional)

You can update your rating any time.

Theory: Line And Curve Intersection Finder

An intersection point is any point \((x,y)\) that satisfies both equations at the same time. The key idea is to reduce the system to a single equation in one variable (often a quadratic).

1) Two lines

A line in standard form is \[ Ax + By + C = 0. \] For two lines \[ A_1x + B_1y + C_1 = 0,\qquad A_2x + B_2y + C_2 = 0, \] we solve a 2×2 linear system. The determinant \[ \Delta = A_1B_2 - A_2B_1 \] tells us what happens:

\(\Delta \neq 0\): one unique intersection point.
\(\Delta = 0\) and not proportional: lines are parallel (no intersection).
\(\Delta = 0\) and proportional: lines are coincident (infinitely many intersections).

2) Line & circle

A common circle form is \[ (x-h)^2 + (y-k)^2 = r^2. \] Expanding gives the general circle form \[ x^2 + y^2 + Dx + Ey + F = 0, \] where \(D=-2h\), \(E=-2k\), \(F=h^2+k^2-r^2\).

If the line is not vertical, write it as \(y=mx+b\) and substitute into the circle equation: \[ x^2 + (mx+b)^2 + Dx + E(mx+b) + F = 0. \] This becomes a quadratic in \(x\), so there can be:

0 real solutions → no intersection
1 real solution → tangent (touches at one point)
2 real solutions → two intersection points

If the line is vertical \(x=x_0\), substitute \(x_0\) into the circle equation to get a quadratic in \(y\).

3) Line & parabola

For an upright parabola \[ y=ax^2+bx+c, \] and a non-vertical line \(y=mx+q\), set them equal: \[ ax^2+bx+c = mx+q \;\;\Rightarrow\;\; ax^2 + (b-m)x + (c-q)=0. \] Again, the intersection points come from solving a quadratic in \(x\), then plugging into \(y=mx+q\).

Graph note

The plot uses square units (equal scaling in x and y) so circles look round and angles/distances are not distorted. Tick labels are drawn near the axes to stay readable while panning/zooming.

Frequently Asked Questions

How do you find the intersection of two lines?

You solve the two line equations as a 2x2 system. Using standard form A1 x + B1 y + C1 = 0 and A2 x + B2 y + C2 = 0, the determinant Delta = A1 B2 - A2 B1 indicates whether there is one intersection (Delta != 0) or the lines are parallel/coincident (Delta = 0).

How do you find the intersection of a line and a circle?

Rewrite the line in a form you can substitute into the circle equation, then reduce to a quadratic in one variable. The number of real solutions determines whether there are 0 intersections, 1 intersection (tangent), or 2 intersections.

How do you find the intersection of a line and y = a x^2 + b x + c?

Set the line equation equal to the parabola equation and rearrange to get a quadratic in x. Solve for x, then substitute each x back into the line (or parabola) to get the corresponding y values.

What does it mean if two lines have no intersection or infinitely many intersections?

No intersection means the lines are parallel (same direction, different position). Infinitely many intersections means the lines are coincident, so they represent the same line.

Rate this calculator

Frequently Asked Questions

Related calculators