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Analytic Geometry: (lesson 1 of 3)

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

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

An equation which can be written in the following form represents a circle except when

D^{2} + E^{2} ≤ 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
x^{2} + y^{2} - 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

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:

Let A(x_{1}, y_{1}) be a point of the circle
(x - a)^{2} + (y - b)^{2} = r^{2} , 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, x_{1} = 4, y_{1} = 6

The equation of tangent is: