# Distances

### The distance between two points

The formula for the distance between two points is obtained from the Pythagorean theorem and is

### The distance between two circles

The distance between two circles is obtained by calculating the distance between the centres and subtracting the radii from it.

Example 1

Calculate the distance between the circles below.

In centre point form

The centre of the first circle is (1,1) and the radius is 3. The centre of the second circle is (10,13) and the radius is 5. Now we can find the distance between the centres.

The distance between the circles is 15-3-5 = 7

If the answer is a negative number, it would mean that the circles intersect. If the distance is zero, the circles are side by side.

### The distance of a point from a line

The distance of a point from a line means the shortest distance of a point from a line. The line segment drawn from the point to the line is perpendicular to the line. The distance is obtained using the formula.

In the formula, in the numerator inside the absolute values is a line in standard form and x₀ and y₀ denote the point whose distance we are finding.

Example 2

Find the distance of the point (2,3) from the line y = -2x + 4

Solution

Let's change the line equation to standard form

2x + y - 4 = 0 and substitute this in the formula.

### The distance between a line and a circle

When finding the distance of a line from a circle, we find the distance of the centre of the circle from the line and subtract the radius of the circle from this.

Example 3

Below are the equations of a line and a circle. Find the distance of the line from the circle.

The line equation in standard form

The circle equation in centre point form

The centre of the circle is (2,0) and the radius is 2. The distance of the centre from the line

The distance of the line from the circle is

If the distance is negative, the line intersects at two points of the circle. When the distance is 0, the line is at a tangent to the circle.

Turn on the subtitles if needed