Circle basics

Circle basics: sample questions

For each of the following equations, state whether it could represent a circle and if so, state the radius and centre.

Question 1

2x^2 + 2y^2 + 4x - 3y - 6 = 0

The Solution

Step 1: x^2 + y^2 + 2x - {3 \over 2}y - 3 = 0

Step 2: g^2 + f^2 - c = 1^2 + ( - {3 \over 4})^2 - ( - 3) = {{73} \over {16}}

The Answer

so equation represents a circle with centre {\rm{( - 1, }}{3 \over 4}), radius {{\sqrt {73} } \over 4}

Question 2

x^2 + y^2 + 2x - 4y + 6 = 0

The Solution

g^2 + f^2 - c = (1)^2 + ( - 2)^2 - 6 = - 1

The Answer

so equation does not represent a circle

When you are trying to build up the equation of the circle, and you know the radius and the centre, it's easier to use the equation (x - a)^2 + (y - b)^2 = r^2, where (a,b) represents the centre of the circle, and r is the radius. This equation is the same as the general equation of a circle, it's just written in a different form.

Follow the worked examples to see how this works.
Write down the equation of the circle with centre (2, -3) and radius \sqrt 7.
\left( {x - 2} \right)^2 + \left( {y - ( - 3)} \right)^2 = \left( {\sqrt 7 } \right)^2
(x - 2)^2 + (y + 3)^2 = 7
If required for further work you can expand this to give
x^2 - 4x + 4 + y^2 + 6y + 9 - 7 = 0
x^2 + y^2 - 4x + 6y + 6 = 0

Try this!
Write down the equation of the circle for each of the following, writing your answer in the form shown in the example.

Question 3

centre = (1,2) radius = \sqrt 5

The Solution

Step 1: (x - 1)^2 + (y - 2)^2 = \left( {\sqrt 5 } \right)^2

Step 2: x^2 - 2x + 1 + y^2 - 4y + 4 = 5

The Answer

x^2 + y^2 - 2x - 4y = 0

Question 4

centre = (0,0) radius = 4

The Solution

(x - 0)^2 + (y - 0)^2 = \left( 4 \right)^2

The Answer

x^2 + y^2 = 16

You'll find both these equations on the formulae sheet in the exam, but you have to know how and when to use them. Here's a quick reminder:

x^2 + y^2 + 2gx + 2fy + c = 0 is used to work out the centre of the circle, and the radius.
(x - a)^2 + (y - b)^2 = r^2 to write the equation of the circle when you know the centre and the radius.

DASAVATARAM

DASAVATARAM