Mathematics • Year 9 • Unit 2 • Lesson 12
Circles — $x^2 + y^2 = r^2$
Build the three-step ritual for the unit-on-the-origin circle: find $r$ by square-rooting, plot the four axis points, draw a smooth circle. Then turn it around — given a radius, write the equation.
1. I do — fully worked example
Read every line. The trap to avoid is using the right-side number as the radius — it is $r^2$, not $r$.
Problem. State the centre and radius of $x^2 + y^2 = 49$, then list the four axis crossings.
Step 1 — Recognise the form.
Equation matches $x^2 + y^2 = r^2$: both $x^2$ and $y^2$ summed with a positive number on the right. So this IS a circle centred at the origin.
Reason: no $h$ or $k$ shift terms appear — centre is $(0, 0)$.
Step 2 — Find $r$ by square-rooting the right side.
$r^2 = 49 \Rightarrow r = \sqrt{49} = 7$.
Reason: the right side is the radius SQUARED, not the radius itself. You must take the square root.
Step 3 — Find the axis crossings.
A circle of radius $r$ centred at the origin crosses the axes at $(\pm r, 0)$ and $(0, \pm r)$.
Reason: at $y = 0$, the equation gives $x = \pm r$. At $x = 0$, it gives $y = \pm r$.
Answer: Centre $(0, 0)$, $r = 7$, axis crossings $(7, 0)$, $(-7, 0)$, $(0, 7)$, $(0, -7)$.
2. We do — fill in the missing steps
Same structure as Section 1. Fill in each blank. 4 marks
Problem. State the centre and radius of $x^2 + y^2 = 100$, then list the four axis crossings.
Step 1 — Form check: the equation looks like $x^2 + y^2 = $ __________________ . That matches the circle pattern with centre at $($ __________________ $)$.
Step 2 — Find $r$: $r^2 = 100$, so $r = \sqrt{\_\_\_\_\_\_\_\_\_\_} = $ __________________ .
Step 3 — Axis crossings: $(r, 0) = ($ ____ $, 0)$; $(-r, 0) = ($ ____ $, 0)$; $(0, r) = (0,$ ____ $)$; $(0, -r) = (0,$ ____ $)$.
Step 4 — Reverse check: sub the point $(0, r)$ into $x^2 + y^2$. We get $0 + $ __________________ $= 100$. Matches $r^2$. So the point lies on the circle.
3. You do — independent practice
Show your working under each problem. 3.1–3.4 are foundation (read $r$ from an equation). 3.5–3.6 are standard (build equation from radius and test a point). 3.7–3.8 are extension (combine point-on-circle reasoning).
Foundation — read the radius
3.1 State the radius of $x^2 + y^2 = 81$. 1 mark
3.2 State the radius of $x^2 + y^2 = 16$. 1 mark
3.3 List the four axis crossings of the circle $x^2 + y^2 = 36$. 1 mark
3.4 Decide whether $x^2 + y^2 = -9$ is a circle. Explain in one sentence. 1 mark
Standard — write equations and test points
3.5 Write the equation of the circle centred at the origin with radius (a) $3$, (b) $10$, (c) $\sqrt{20}$. 2 marks
3.6 For the circle $x^2 + y^2 = 25$, decide whether each point lies on, inside, or outside the circle: (a) $(3, 4)$ (b) $(2, 2)$ (c) $(0, 6)$. Show your substitution working. 2 marks
Extension — circle from a point
3.7 The point $(5, 12)$ lies on a circle centred at the origin. Find $r$ and write the equation of the circle. 2 marks
3.8 Two circles are centred at the origin. Circle A has radius $5$. Circle B passes through $(6, 8)$. (a) Find the equation of each. (b) Which circle is bigger? Justify with the two $r$ values. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (faded $x^2 + y^2 = 100$)
Step 1: $x^2 + y^2 = \mathbf{100}$, centre $\mathbf{(0, 0)}$.
Step 2: $r = \sqrt{\mathbf{100}} = \mathbf{10}$.
Step 3: $(\mathbf{10}, 0)$, $(\mathbf{-10}, 0)$, $(0, \mathbf{10})$, $(0, \mathbf{-10})$.
Step 4: $0 + \mathbf{100} = 100 = r^2$. The point $(0, 10)$ is ON the circle.
3.1 — $x^2 + y^2 = 81$
$r^2 = 81 \Rightarrow r = \sqrt{81} = \mathbf{9}$.
3.2 — $x^2 + y^2 = 16$
$r^2 = 16 \Rightarrow r = \sqrt{16} = \mathbf{4}$.
3.3 — Axis crossings of $x^2 + y^2 = 36$
$r = 6$. Crossings: $(6, 0)$, $(-6, 0)$, $(0, 6)$, $(0, -6)$.
3.4 — $x^2 + y^2 = -9$
NOT a circle. The left side $x^2 + y^2$ is always $\ge 0$ for real $x$ and $y$, so it can never equal $-9$. No real points satisfy the equation.
3.5 — Equations from radius
(a) $r = 3 \Rightarrow r^2 = 9$: $\mathbf{x^2 + y^2 = 9}$.
(b) $r = 10 \Rightarrow r^2 = 100$: $\mathbf{x^2 + y^2 = 100}$.
(c) $r = \sqrt{20} \Rightarrow r^2 = 20$: $\mathbf{x^2 + y^2 = 20}$.
3.6 — On, inside, outside ($r^2 = 25$)
(a) $3^2 + 4^2 = 9 + 16 = 25 = r^2$ — ON the circle.
(b) $2^2 + 2^2 = 4 + 4 = 8 < 25$ — INSIDE.
(c) $0^2 + 6^2 = 36 > 25$ — OUTSIDE.
3.7 — Circle through $(5, 12)$
$r^2 = 5^2 + 12^2 = 25 + 144 = 169$. So $r = \sqrt{169} = 13$. Equation: $\mathbf{x^2 + y^2 = 169}$.
3.8 — Circle A vs Circle B
(a) Circle A: $r = 5 \Rightarrow r^2 = 25$, so $\mathbf{x^2 + y^2 = 25}$.
Circle B passes through $(6, 8)$, so $r^2 = 36 + 64 = 100$, giving $r = 10$ and $\mathbf{x^2 + y^2 = 100}$.
(b) Circle B has $r = 10$ vs Circle A's $r = 5$, so Circle B is bigger (double the radius).