**Summary**

Formula for equation of a circle is:

Where *(a, b)* is the center of the circle and *r* is the radius.

Before we move on to the equation of a circle, let’s just recap Pythagoras theorem from your GCSE studies:

We know that Pythagoras Theorem can be applied when we have a right angle triangle with a missing length of any side.

i.e refer to Fig 1 where:

*a* is the length of the triangle,

*b* is the width of the triangle &

*c* is the hypotenuse

The formula for Pythagoras theorem is:

Pythagoras theorem plays a key role in deriving the equation of a circle.

#### The equation of a Circle

Look at the circle in Fig 2.

The centre of the circle is *Q (4, 6)*.

The radius of the circle is *r* which is the length from* Q to P* where point P is (*x, y*).

We can find the value of r using the pythagorean theorem as a right angle triangle is formed with height *n* and width *m*:

We can see that lengths and .

Now substituting the values in the formula above we get:

*This is the equation of the circle in Fig 2*

In general, a circle with radius* r* and center (*a, b*) has the equation:

#### Example #1

Q. What is the center and radius of the circle with equation:

*Solution:*

From the equation we can see that the center of the circle is:

*(a, b) = (5, 7)*

And the radius of the circle is:

#### Example #2

Q. Write down the equation of the circle in Fig 3.

*Solution:*

From the diagram, we can see that center of the circle is *Q(8, 6)* and the radius is *r = 2*

Putting the above values in the general formula for equation of a circle, we get:

**Ans:**

##### Reference:

CGP AS-level mathematics edexcel revision and practice book