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Determine a circle's equation and features

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What should be done first with the displayed circle equation?

\[\left(x-4\right)^2+\left(y+3\right)^2=25\]
Year 11, VCE Units 1 & 2, Chapter 10, 4 difficulty levels, Calculator allowedBuild a worksheet

Circle equation and features

A circle in centre-radius form gives its centre and squared radius directly after the signs are interpreted correctly. The centre coordinates and positive radius also determine the domain and range

\[\left(x-h\right)^2+\left(y-k\right)^2=r^2\]
  1. Read opposite centre signs The signs inside the brackets are opposite to the centre coordinates \(\displaystyle \left(x-4\right)^2+\left(y+3\right)^2=25\Rightarrow C=(4,-3)\)
  2. Take the positive radius The right side is the squared radius, so take its positive square root \(\displaystyle r^2=49\Rightarrow r=7\)
  3. Normalise before completing squares Divide by a common coefficient of the squared terms before reading the centre \(\displaystyle 2x^2+2y^2-8x+12y=18\Rightarrow x^2+y^2-4x+6y=9\)
  4. Use extrema for domain and range Add and subtract the radius from the matching centre coordinate \(\displaystyle C=(4,-3),\ r=7\Rightarrow\operatorname{Dom}=[-3,11],\quad\operatorname{Ran}=[-10,4]\)

Worked example

A circle has the centre and positive radius shown below. State its centre-radius equation, domain and range

\[C=(-3,2),\quad r=4\]
  1. \(\displaystyle h=-3,\qquad k=2,\qquad r^2=4^2=16\)Record the centre coordinates and square the positive radius
  2. \(\displaystyle \left(x+3\right)^2+\left(y-2\right)^2=16\)Use opposite signs inside the brackets and place the squared radius on the right
  3. \(\displaystyle \operatorname{Dom}=[-3-4,-3+4]=[-7,1]\)The domain uses the horizontal centre coordinate and the radius
  4. \(\displaystyle \operatorname{Ran}=[2-4,2+4]=[-2,6]\)The range uses the vertical centre coordinate and the radius

Common mistakes

Centre signs are copied
✕\(\displaystyle \left(x+2\right)^2+\left(y-5\right)^2=16\Rightarrow C=(2,-5)\)✓\(\displaystyle \left(x+2\right)^2+\left(y-5\right)^2=16\Rightarrow C=(-2,5)\)
The signs inside the brackets are opposite to the centre coordinates
Squared radius is called the radius
✕\(\displaystyle r^2=25\Rightarrow r=25\)✓\(\displaystyle r^2=25\Rightarrow r=\sqrt{25}=5\)
The right side is the squared radius, so the radius is its positive square root
The full linear coefficient completes a square
✕\(\displaystyle x^2+6x+y^2-4y=12\Rightarrow\left(x+6\right)^2+\left(y-4\right)^2=64\)✓\(\displaystyle x^2+6x+y^2-4y=12\Rightarrow\left(x+3\right)^2+\left(y-2\right)^2=25\)
Use half of each linear coefficient when completing a square

Difficulty levels

introRead the centre and a rational radius from centre-radius form
standardForm a circle equation from geometric data and state its domain and range
challengeNormalise an expanded relation and complete both squares exactly
examSelect correct circle features and equivalent equations from diagnostic alternatives

Questions about this topic

Why are the centre signs opposite to the bracket signs?+

The centre coordinates have the opposite signs to the constants inside their brackets. Adding 2 to x gives a horizontal centre coordinate of negative 2; subtracting 5 from y gives a vertical centre coordinate of 5

Why is the radius positive?+

A radius is a length, so it is positive. The equation stores the squared radius on the right side

How are the domain and range found from a circle?+

Add and subtract the radius from the horizontal centre coordinate for the domain and from the vertical centre coordinate for the range

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Specialist Mathematics, chapter 10: Extended functions and relations

In this chapter

Decompose rational expressions into partial fractionsDetermine parameters in a polynomial identityReciprocal and inverse circular functionsConstruct conic loci and convert coordinate formsDetermine a circle's equation and featuresTransform absolute-value graphs and find intersections