Analytic Geometry Calculator

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equation of a line through two points

Equation of a line through two points

In the Line equation tab, keep the Two points option selected and enter A and B. The cards give the four forms of the equation, where the line crosses each axis and the direction vector, with a graph of the line and its intercepts.

Two routes to the same line and how to move between forms

The most direct route uses the slope: m = (y₂ - y₁)/(x₂ - x₁) and then y - y₁ = m(x - x₁). With A(1, 2) and B(5, 4), m = 2/4 = 1/2, so y - 2 = (1/2)(x - 1) and, solving for y, y = (1/2)x + 3/2. The second route is the determinant of x, y, 1 with the two points set to zero, which gives (y₁ - y₂)x + (x₂ - x₁)y + (x₁y₂ - x₂y₁) = 0 without dividing by anything. It works even when x₁ = x₂, the case where the first route stalls, because m would be a division by zero.
The standard form ax + by + c = 0 is written here with integer coefficients, no common factor and a positive, so that two people get the same answer: x - 2y + 3 = 0 rather than 2x - 4y + 6 = 0 or -x + 2y - 3 = 0. The other forms follow from it: m = -a/b and n = -c/b for slope-intercept, and p = -c/a and q = -c/b for the intercept form x/p + y/q = 1. The intercept form exists only when a, b and c are all nonzero, that is, when the line misses the origin and is not parallel to an axis. For the example, p = -3 and q = 3/2.
The special cases deserve attention. A horizontal line (y₁ = y₂) has m = 0 and equation y = k. A vertical line (x₁ = x₂) has undefined m, a 90° angle and equation x = k, with no slope-intercept form. A line through the origin has c = 0, so it has no intercept form. The parametric form uses A as the starting point and B - A as the direction vector: for the example, x = 1 + 4t and y = 2 + 2t, where t = 0 gives A and t = 1 gives B. The angle of inclination α = arctan(m) is 26.5651° for m = 1/2 and becomes obtuse when m is negative.

Checked examples

Fractional slope

Input: A(1, 2), B(5, 4)
Expected output: x - 2y + 3 = 0; y = (1/2)x + 3/2

m = 1/2, α ≈ 26.5651°.

Passes through (0, 1)

Input: A(0, 1), B(2, 5)
Expected output: 2x - y + 1 = 0; y = 2x + 1

n = 1 is where it crosses the y-axis.

Vertical line

Input: A(3, 1), B(3, 9)
Expected output: x - 3 = 0; x = 3

m undefined, α = 90°.

Horizontal line

Input: A(-1, 4), B(5, 4)
Expected output: y - 4 = 0; y = 4

m = 0, α = 0°.

Negative slope

Input: A(2, -3), B(-2, 1)
Expected output: x + y + 1 = 0; y = -x - 1

m = -1, α = 135°.

Intercept form

Input: A(1, 0), B(0, 2)
Expected output: 2x + y - 2 = 0; x/1 + y/2 = 1

Crosses the axes at 1 and 2.

Full tool FAQ

How do you calculate the distance between two points?

Use d = √((x₂ - x₁)² + (y₂ - y₁)²), which is the Pythagorean theorem applied to the coordinate differences. Between A(1, 2) and B(5, 4): d = √(4² + 2²) = √20 = 2√5 ≈ 4.4721. The calculator simplifies the radical by pulling perfect squares out of the root (20 = 4 · 5) and shows the decimal next to it.

How do you find the equation of the line through two points?

What is the difference between the standard, slope-intercept, intercept and parametric forms?

How do you calculate the slope and the angle of inclination?

How can you tell whether two lines are parallel or perpendicular?

How do you find the intersection of two lines?

How do you calculate the distance from a point to a line?

How do you check whether three points are collinear and find the triangle area?

How do you get the equation of a circle and go from general form back to center form?

Are the results exact? What are the limits?

Frequently asked questions

Which form of the line equation should I give on an exam?

Use the one the question asks for. If it does not say, the standard form (ax + by + c = 0) is the safest because it exists for every line, vertical ones included. The slope-intercept form is good for reading m and n and for sketching.

Why does my equation differ from the answer key?

Multiplying the whole equation by a nonzero constant describes the same line. 2x - 4y + 6 = 0 and x - 2y + 3 = 0 are equal. Compare coefficients after dividing by their greatest common divisor, or check that both points satisfy your equation.

How do I find out whether a third point is on the line?

Substitute its coordinates into the standard form: if ax + by + c comes out zero, the point is on the line. Another option is to type the three points in the Triangle tab and see whether the determinant is zero.

Next steps

Two routes to the same line and how to move between forms

The most direct route uses the slope: m = (y₂ - y₁)/(x₂ - x₁) and then y - y₁ = m(x - x₁). With A(1, 2) and B(5, 4), m = 2/4 = 1/2, so y - 2 = (1/2)(x - 1) and, solving for y, y = (1/2)x + 3/2. The second route is the determinant of x, y, 1 with the two points set to zero, which gives (y₁ - y₂)x + (x₂ - x₁)y + (x₁y₂ - x₂y₁) = 0 without dividing by anything. It works even when x₁ = x₂, the case where the first route stalls, because m would be a division by zero.

The standard form ax + by + c = 0 is written here with integer coefficients, no common factor and a positive, so that two people get the same answer: x - 2y + 3 = 0 rather than 2x - 4y + 6 = 0 or -x + 2y - 3 = 0. The other forms follow from it: m = -a/b and n = -c/b for slope-intercept, and p = -c/a and q = -c/b for the intercept form x/p + y/q = 1. The intercept form exists only when a, b and c are all nonzero, that is, when the line misses the origin and is not parallel to an axis. For the example, p = -3 and q = 3/2.

The special cases deserve attention. A horizontal line (y₁ = y₂) has m = 0 and equation y = k. A vertical line (x₁ = x₂) has undefined m, a 90° angle and equation x = k, with no slope-intercept form. A line through the origin has c = 0, so it has no intercept form. The parametric form uses A as the starting point and B - A as the direction vector: for the example, x = 1 + 4t and y = 2 + 2t, where t = 0 gives A and t = 1 gives B. The angle of inclination α = arctan(m) is 26.5651° for m = 1/2 and becomes obtuse when m is negative.

Checked examples

Fractional slope

Input: A(1, 2), B(5, 4)
Expected output: x - 2y + 3 = 0; y = (1/2)x + 3/2

m = 1/2, α ≈ 26.5651°.

Passes through (0, 1)

Input: A(0, 1), B(2, 5)
Expected output: 2x - y + 1 = 0; y = 2x + 1

n = 1 is where it crosses the y-axis.

Vertical line

Input: A(3, 1), B(3, 9)
Expected output: x - 3 = 0; x = 3

m undefined, α = 90°.

Horizontal line

Input: A(-1, 4), B(5, 4)
Expected output: y - 4 = 0; y = 4

m = 0, α = 0°.

Negative slope

Input: A(2, -3), B(-2, 1)
Expected output: x + y + 1 = 0; y = -x - 1

m = -1, α = 135°.

Intercept form

Input: A(1, 0), B(0, 2)
Expected output: 2x + y - 2 = 0; x/1 + y/2 = 1

Crosses the axes at 1 and 2.

Full tool FAQ

How do you calculate the distance between two points?

How do you find the equation of the line through two points?

What is the difference between the standard, slope-intercept, intercept and parametric forms?

How do you calculate the slope and the angle of inclination?

How can you tell whether two lines are parallel or perpendicular?

How do you find the intersection of two lines?

How do you calculate the distance from a point to a line?

How do you check whether three points are collinear and find the triangle area?

How do you get the equation of a circle and go from general form back to center form?