Ellipse »

Definition:

The General Form of the Circle

An equation which can be written in the following form represents a circle except when
D² + E² ≤ F

This is called the general form of the circle .

is the centre of the circle and the radius is

Example 1:

Find the radius and centre of the circle x² + y² - 2x - 4y + 1 = 0

Solution:

We need to get the equation into the form:

The radius of circle is r = 2 and the centre of the circle is O(1, 2).

In this example D = -1, E = 2, F = 1

Equation of a Circle from 3 Points

Example 2:

Find the equation of the circle through the points A(4, -2), B(6, 1), C(-1, 3).

Let represent the circle. Then, since A is on the circle, its coordinates, 4 and -2, satisfy the equation

Whence: 8D - 4E + F = -20.

Similary, for B, 12D + 2E + F = -37.

and for C, -2D + 6E + F = -10.

Solving, we have , and the equation is:

Equation of a tangent at a given point

Let A(x₁, y₁) be a point of the circle (x - a)² + (y - b)² = r² , then an equation of tangent line to circle is:

Example:

Given the circle (x - 1)² + (y - 2)² = 25 and the point A(4,6) on the circle. Find the equation of the tangent to the circle at A.

Solution:

Here we have: a = 1, b = 2, x₁ = 4, y₁ = 6

The equation of tangent is: