Chapter 05: Circles – Part 2: Completing the Square and Finding Circle Equations

CHAPTER 05: CIRCLES

PART 2: COMPLETING THE SQUARE AND FINDING CIRCLE EQUATIONS

A-Level Pure Mathematics | Mathematics HipHop Academy

---

1. Learning Objectives

By the end of this lesson, you should be able to:

• Complete the square for quadratic expressions in \(x\) and \(y\).
• Convert an expanded circle equation into standard form.
• Find the centre and radius by completing the square.
• Determine the equation of a circle from its centre and a point on its circumference.
• Find a circle equation when the endpoints of a diameter are given.
• Determine the equation of a circle passing through three given points.
• Use given conditions to find unknown constants in a circle equation.

2. Revision: The Two Forms of a Circle Equation

In Part 1, we studied the standard equation:

\((x-a)^2+(y-b)^2=r^2\)

Centre: \((a,b)\)

Radius: \(r\)

We also studied the expanded or general form:

\(x^2+y^2+2gx+2fy+c=0\)

Centre: \((-g,-f)\)

Radius: \(\sqrt{g^2+f^2-c}\)

Completing the square allows us to move from the expanded form to the standard form.

3. Understanding Completing the Square

Completing the square is a method of rewriting a quadratic expression as the square of a bracket, together with a constant.

Consider:

\(x^2+6x\)

Take half the coefficient of \(x\):

\(\frac{6}{2}=3\)

Square this number:

\(3^2=9\)

Add and subtract 9:

\(x^2+6x+9-9\)

Factorise the first three terms:

\((x+3)^2-9\)

General rule

For an expression \(x^2+px\):

\(x^2+px=\left(x+\frac p2\right)^2-\frac{p^2}{4}\)

In simple terms:

1. Take half the coefficient of the variable.
2. Square the result.
3. Add that square inside the expression.
4. Subtract the same square to keep the expression unchanged.

4. Completing the Square in Two Variables

A circle equation contains both \(x\) and \(y\). We complete the square separately for each variable.

For example:

\(x^2+6x+y^2-8y\)

Group the \(x\)-terms and \(y\)-terms:

\((x^2+6x)+(y^2-8y)\)

Complete the square for \(x\):

\(x^2+6x=(x+3)^2-9\)

Complete the square for \(y\):

\(y^2-8y=(y-4)^2-16\)

Therefore:

\((x+3)^2-9+(y-4)^2-16\)

5. Converting an Expanded Circle Equation into Standard Form

Consider the equation:

\(x^2+y^2+6x-8y-11=0\)

Step 1: Group the \(x\)-terms and \(y\)-terms, moving the constant to the right.

\((x^2+6x)+(y^2-8y)=11\)

Step 2: Complete the square for each group.

\(x^2+6x=(x+3)^2-9\)

\(y^2-8y=(y-4)^2-16\)

Step 3: Substitute:

\((x+3)^2-9+(y-4)^2-16=11\)

Step 4: Collect the constants on the right:

\((x+3)^2+(y-4)^2=11+9+16\)

STANDARD FORM

\((x+3)^2+(y-4)^2=36\)

Centre: \((-3,4)\)

Radius: \(6\)

6. A General Method for Finding the Centre and Radius

Given an equation in the form:

\(x^2+y^2+Dx+Ey+F=0\)

Follow these steps:

1. Group the \(x\)-terms and \(y\)-terms.
2. Move the constant term to the right-hand side.
3. Complete the square for the \(x\)-terms.
4. Complete the square for the \(y\)-terms.
5. Simplify to obtain standard form.
6. Read off the centre and radius.

Remember that the coefficients of \(x^2\) and \(y^2\) must be equal and there must be no \(xy\) term for this simple circle form.

7. Worked Examples

Example 1: Finding the Centre and Radius by Completing the Square

Find the centre and radius of:

\(x^2+y^2-4x+10y+13=0\)

Solution

Group the terms:

\((x^2-4x)+(y^2+10y)=-13\)

Complete the square:

\(x^2-4x=(x-2)^2-4\)

\(y^2+10y=(y+5)^2-25\)

Substitute:

\((x-2)^2-4+(y+5)^2-25=-13\)

Rearrange:

\((x-2)^2+(y+5)^2=-13+4+25\)

\((x-2)^2+(y+5)^2=16\)

Answer: Centre = (2, -5), Radius = 4 units.

Example 2: Converting an Equation with a Positive Constant

Express the equation below in standard form and state its centre and radius:

\(x^2+y^2+8x+2y+8=0\)

Solution

Group the terms:

\((x^2+8x)+(y^2+2y)=-8\)

Complete the square:

\(x^2+8x=(x+4)^2-16\)

\(y^2+2y=(y+1)^2-1\)

Substitute:

\((x+4)^2-16+(y+1)^2-1=-8\)

Rearrange:

\((x+4)^2+(y+1)^2=9\)

Answer: Centre = (-4, -1), Radius = 3 units.

Example 3: Finding the Equation from the Centre and a Point

A circle has centre \(C(3,2)\) and passes through \(P(6,6)\). Find its equation in both standard and expanded forms.

Solution

First find the radius:

\(r=\sqrt{(6-3)^2+(6-2)^2}\)

\(r=\sqrt{9+16}=\sqrt{25}=5\)

Use the standard equation:

\((x-3)^2+(y-2)^2=25\)

Expand:

\(x^2-6x+9+y^2-4y+4=25\)

Collect terms:

Standard form: \((x-3)^2+(y-2)^2=25\)

Expanded form: \(x^2+y^2-6x-4y-12=0\)

Example 4: Finding a Circle Equation from the Endpoints of a Diameter

The endpoints of a diameter of a circle are \(A(2,1)\) and \(B(8,5)\). Find the equation of the circle.

Solution

The centre is the midpoint of the diameter:

\(C=\left(\frac{2+8}{2},\frac{1+5}{2}\right)\)

\(C=(5,3)\)

Find the diameter:

\(AB=\sqrt{(8-2)^2+(5-1)^2}\)

\(=\sqrt{36+16}=\sqrt{52}=2\sqrt{13}\)

Therefore the radius is:

\(r=\sqrt{13}\)

Substitute into the standard equation:

Answer: \((x-5)^2+(y-3)^2=13\)

Example 5: Finding the Equation of a Circle Through Three Points

Find the equation of the circle passing through:

\(A(0,0),\quad B(4,0),\quad C(0,6)\)

Solution

Start with the general equation:

\(x^2+y^2+2gx+2fy+c=0\)

Since \(A(0,0)\) lies on the circle:

\(0+0+0+0+c=0\), so \(c=0\).

Substitute \(B(4,0)\):

\(16+8g=0\)

\(8g=-16\), therefore \(g=-2\).

Substitute \(C(0,6)\):

\(36+12f=0\)

\(12f=-36\), therefore \(f=-3\).

Substitute the values of \(g\), \(f\) and \(c\):

\(x^2+y^2-4x-6y=0\)

To find the centre and radius, complete the square:

\((x-2)^2-4+(y-3)^2-9=0\)

\((x-2)^2+(y-3)^2=13\)

Answer: \(x^2+y^2-4x-6y=0\)

Centre: \((2,3)\), Radius: \(\sqrt{13}\).

8. Finding Unknown Constants Using Given Conditions

Sometimes a circle equation contains unknown constants. We use the information given in the question to determine those constants.

For example, if a circle has equation:

\(x^2+y^2+2gx+2fy+c=0\)

and a point \((p,q)\) lies on the circle, substitute \(x=p\) and \(y=q\):

\(p^2+q^2+2gp+2fq+c=0\)

Each point on the circumference gives an equation. These equations can be solved simultaneously when enough independent information is available.

Example 6: Using a Point to Find an Unknown Constant

A circle has equation:

\(x^2+y^2+6x+2y+k=0\)

If the point \((1,2)\) lies on the circle, find \(k\).

Solution

Substitute \(x=1\) and \(y=2\):

\(1^2+2^2+6(1)+2(2)+k=0\)

\(1+4+6+4+k=0\)

\(15+k=0\)

Answer: \(k=-15\).

9. Conditions for a Valid Circle Equation

For an equation in the form:

\(x^2+y^2+2gx+2fy+c=0\)

the radius is given by:

\(r=\sqrt{g^2+f^2-c}\)

Therefore:

• If \(g^2+f^2-c>0\), the equation represents a real circle with positive radius.
• If \(g^2+f^2-c=0\), the equation represents a single point.
• If \(g^2+f^2-c<0\), there is no real circle.

When completing the square, always check that the final value of \(r^2\) is sensible.

10. Common Examination Mistakes

• Adding the square of half the coefficient without subtracting it elsewhere.
• Using the wrong sign when identifying the centre.
• Forgetting to move the constant to the right-hand side before completing the square.
• Calculating the diameter but using it as the radius.
• Substituting only one coordinate of a point into a circle equation.
• Failing to check that all given points satisfy the final equation.

11. Practice Exercises

Attempt the questions independently before opening the answer sections.

Exercise A: Completing the Square

1. Express \(x^2+y^2+4x-6y-12=0\) in standard form. State its centre and radius.
2. Express \(x^2+y^2-8x+2y+8=0\) in standard form. Find its centre and radius.
3. Find the centre and radius of \(x^2+y^2+10x+12y+36=0\).
4. Convert \(x^2+y^2-2x-4y-20=0\) into standard form.

Exercise B: Constructing Circle Equations

5. Find the equation of a circle with centre \((-2,3)\) passing through \((1,7)\).
6. The endpoints of a diameter are \((-1,2)\) and \((5,6)\). Find the centre, radius and equation of the circle.
7. Find the equation of the circle passing through \(A(0,0)\), \(B(6,0)\) and \(C(0,8)\).
8. A circle has centre \((4,-1)\) and radius \(\sqrt{20}\). Write its equation in standard form and expanded form.

Exercise C: Examination Challenge

9. The circle \(x^2+y^2+2gx+2fy+c=0\) has centre \((3,-2)\) and passes through \((6,2)\). Find \(g\), \(f\), \(c\) and its equation.
10. The equation \(x^2+y^2-6x+4y+k=0\) represents a circle. Find the range of values of \(k\) for which the circle has a positive radius.
11. A circle passes through \((1,1)\), \((5,1)\) and \((1,5)\). Find its equation, centre and radius.
12. A circle has centre on the \(x\)-axis and passes through \((2,3)\) and \((6,3)\). Find its centre and equation.

OPEN ANSWERS – EXERCISE A

1. \((x+2)^2+(y-3)^2=25\). Centre \((-2,3)\), radius \(5\).
2. \((x-4)^2+(y+1)^2=9\). Centre \((4,-1)\), radius \(3\).
3. \((x+5)^2+(y+6)^2=25\). Centre \((-5,-6)\), radius \(5\).
4. \((x-1)^2+(y-2)^2=25\). Centre \((1,2)\), radius \(5\).

OPEN ANSWERS – EXERCISE B

5. Radius \(=5\). Equation: \((x+2)^2+(y-3)^2=25\).
6. Centre \((2,4)\); diameter \(=2\sqrt{13}\); radius \(=\sqrt{13}\). Equation: \((x-2)^2+(y-4)^2=13\).
7. Equation: \(x^2+y^2-6x-8y=0\). Centre \((3,4)\), radius \(5\).
8. Standard form: \((x-4)^2+(y+1)^2=20\). Expanded form: \(x^2+y^2-8x+2y-3=0\).

OPEN ANSWERS – EXERCISE C

9. Centre \((3,-2)\) gives \(g=-3\), \(f=2\). Using \((6,2)\): \(36+4-36+8+c=0\), so \(c=-12\). Equation: \(x^2+y^2-6x+4y-12=0\).
10. Centre \((3,-2)\), and \(r^2=9+4-k=13-k\). Positive radius requires \(k<13\).
11. Equation: \(x^2+y^2-6x-6y+10=0\). Centre \((3,3)\), radius \(\sqrt{8}=2\sqrt2\).
12. The centre lies on the perpendicular bisector of the chord joining \((2,3)\) and \((6,3)\). Its midpoint is \((4,3)\), so the centre is \((4,0)\). Radius \(=5\). Equation: \((x-4)^2+y^2=25\).

12. Part 2 Summary

Completing the square:

\(x^2+px=\left(x+\frac p2\right)^2-\frac{p^2}{4}\)

Standard circle equation:

\((x-a)^2+(y-b)^2=r^2\)

Centre: \((a,b)\)

Radius: \(r\)

General circle equation:

\(x^2+y^2+2gx+2fy+c=0\)

Centre in general form: \((-g,-f)\)

Radius in general form: \(\sqrt{g^2+f^2-c}\)

Midpoint of a diameter:

\(\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\)

Coming Next: Part 3

In Part 3, we shall study Lines and Circles – Intersections and Tangents. We will investigate how a straight line meets a circle, how to determine the number of intersection points, and how to find tangent equations.

---

MATHEMATICS HIPHOP ACADEMY

Making Mathematics Easier, One Concept at a Time.