# Equation of a Circle

In this topic we will learn how to create an equation of a circle with different given. Being familiar of the method will boost our speed and accuracy. Most of the problems we encounter at school are just recycled.

General equation of a circle:

where C, D and E are real numbers

Equation of a circle in center-radius form:

where (h,k) is the vertex of the circle and *r *is the radius of

the circle

**Sample Problem 1: **(4^{th} Year 2013 MMC Elimination)

Find the center and the radius of the circle

Solution:

To solve this problem we need to convert this standard form to center radius form by completing the square of x and y.

From the equation above we can see that h=2, k=-7 and radius of √6

The desired answer is (2,-7) and units

**Sample Problem 2:** (4^{th} Year 2005 MMC elimination)

Write the equation of the circle tangent to y=-3 and with the center at (-2,2) in center radius form.

Solution:

We write the equation of the circle:

, we are given already with . We just need the radius.

Solve for radius:

The radius of the circle is equal to the distance between the point (-2,2) and y=-3. Since the circle passes line y=-3 it intersects the point the point (-2,-3). It is now obvious that the radius is 5 units.

**Sample Problem 3:** *(4 ^{th} year 2005 MMC elimination)*

Find the equation of the circle in standard form concentric with x^{2}+y^{2}+4x-8y-12=0 and passing through (2,-1)

Solution:

Concentric circle are circles with the same center. We can get the center of the circle from x^{2}+y^{2}+4x-8y-12=0. By completing the square we get h=-2, k=4. To solve for the radius of the desired circle we solve for the distance between the center (-2,4) and a point on the circumference of the circle (2,-1) using the distance formula.

By substitution:

Solving for the equation of the circle: