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:
Q. What is the center and radius of the circle with equation:
From the equation we can see that the center of the circle is:
(a, b) = (5, 7)
And the radius of the circle is:
Q. Write down the equation of the circle in Fig 3.
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:
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