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Analytic Geometry: (lesson 1 of 3)

Circle

Definition:

An equation of circle of radius 'r' with a center in a point Circle Definitions is:

Circle Definitions

If a center of the circle coincides with the origin of coordinates, then an equation of circle is:

Circle Definitions

The General Form of the Circle

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

General form of the circle

This is called the general form of the circle .

General form of the circle is the centre of the circle and the radius is General form of the circle

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: Circle Example 1

Circle Solution 1

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 Equation of a Circle from 3 Points represent the circle. Then, since A is on the circle, its coordinates, 4 and -2, satisfy the equation

Equation of a Circle from 3 Points

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

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

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

Solving, we have Equation of a Circle from 3 Points, and the equation is: Equation of a Circle from 3 Points

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:

Circle Equation of tangent line

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:

Circle Equation of tangent line