Distance from a point to a line

The shortest road to a line

Stand at a point $P$ and walk to a straight road. Which way is shortest? Straight across, at a right angle to the road. The distance from a point to a line is the length of that perpendicular.

The formula

The distance of the point $(x_1, y_1)$ from the line $Ax + By + C = 0$ is

Put the point into the left side of the equation, take the size of the result, and divide by $\sqrt{A^2 + B^2}$.

Example. Find the distance of $(5, 1)$ from $3x + 4y - 10 = 0$.

Step 1. Put the point in. $3(5) + 4(1) - 10 = 9$.

Step 2. Divide. $\sqrt{3^2 + 4^2} = 5$, so $d = \dfrac{9}{5} = 1.8$.

Where the formula comes from

Write the line in normal form: $x\cos\alpha + y\sin\alpha = p$. The line through $P$ parallel to it is $x\cos\alpha + y\sin\alpha = p'$, where $p' = x_1\cos\alpha + y_1\sin\alpha$.

Two parallel lines with the same unit normal $(\cos\alpha, \sin\alpha)$ are $|p' - p|$ apart. So

Now put $\cos\alpha = \dfrac{A}{R}$, $\sin\alpha = \dfrac{B}{R}$ and $p = -\dfrac{C}{R}$, where $R = \sqrt{A^2 + B^2}$ and the signs are chosen as in the normal form. The result is $\dfrac{|Ax_1 + By_1 + C|}{R}$. The size bars cover the choice of sign.

Drag $P$ to the other side of the line. The value of $Ax + By + C$ changes sign, and the distance stays positive.

The distance of the origin

Put $x_1 = y_1 = 0$. The distance of the origin from $Ax + By + C = 0$ is

which is the number $p$ of the normal form.

The distance between parallel lines

Two parallel lines $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$ are

apart. The formula needs the same $A$ and $B$ in both equations.

Example. Find the distance between $3x + 4y - 10 = 0$ and $6x + 8y + 10 = 0$.

Step 1. Match the coefficients. Divide the second equation by $2$: $3x + 4y + 5 = 0$.

Step 2. Apply the formula. $d = \dfrac{|-10 - 5|}{\sqrt{3^2 + 4^2}} = \dfrac{15}{5} = 3$.

A common slip is to use $-10$ and $+10$ with the coefficients $3$ and $4$. That gives $4$, which is wrong, because the second line was never scaled.

Which side of a line?

The expression $Ax + By + C$ is $0$ on the line, positive on one side and negative on the other. Two points are on the same side when it has the same sign at both. They are on opposite sides when the signs differ.

Example. Where do $(3, 1)$ and $(1, 1)$ lie compared with the line $2x - y - 4 = 0$?

Step 1. Test each point. At $(3, 1)$: $6 - 1 - 4 = 1$, positive. At $(1, 1)$: $2 - 1 - 4 = -3$, negative.

Step 2. Compare the signs. The signs differ, so the points are on opposite sides. The segment between them crosses the line.

The origin is a handy test point. Its value is just $C$, so a point is on the origin's side when its value has the same sign as $C$.

Points at a given distance

Example. Find the points on the $x$-axis that are $5$ units from $5x + 12y - 10 = 0$.

Step 1. Write the point. It is $(a, 0)$.

Step 2. Use the formula. $\dfrac{|5a - 10|}{13} = 5$, so $|5a - 10| = 65$.

Step 3. Solve both cases. $5a - 10 = 65$ gives $a = 15$. $5a - 10 = -65$ gives $a = -11$.

Answer: $(15, 0)$ and $(-11, 0)$. The absolute value always gives two cases.

Practise with new numbers

Try it in Python

The program finds the distance twice. First by the formula. Then by walking along the line in steps of $0.001$ and keeping the nearest point.

Common mistakes

• Leaving out the absolute value. A distance is never negative.
• Forgetting the square root. The bottom is $\sqrt{A^2 + B^2}$, not $A^2 + B^2$ and not $A + B$.
• Using the formula for parallel lines with different coefficients. Make $A$ and $B$ the same in both equations first.
• Putting the point into the equation with the signs wrong. The equation must have $0$ on the right, so $3x + 4y = 10$ becomes $3x + 4y - 10$.
• Deciding the side from the size of the value. Only the sign matters.

Remember these

• $d = \dfrac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$, and for the origin $d = \dfrac{|C|}{\sqrt{A^2 + B^2}}$.
• Parallel lines: $d = \dfrac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}$, after the coefficients are matched.
• The sign of $Ax_1 + By_1 + C$ tells the side: the same sign means the same side.
• A point at a given distance leads to $|\ldots| =$ a number, which gives two cases.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Find the distance of the origin from $3x + 4y - 15 = 0$.

2. Find the distance of $(3, -1)$ from $4x - 3y + 6 = 0$.

3. Find the distance between $3x + 4y + 7 = 0$ and $3x + 4y - 3 = 0$.

4. Are $(2, 3)$ and $(-1, 4)$ on the same side of $x - y + 2 = 0$?

5. Find the points on the $y$-axis that are $3$ units from $3x + 4y - 5 = 0$.