Mathematics • Year 9 • Unit 2 • Lesson 12
Circles — Mixed Challenge
Pull together every circle skill: read $r$, build equations from points, decide on/inside/outside, recognise non-circle equations, and split a circle into upper and lower halves. Then catch a mistake and design your own circles.
1. Mixed problems
Each question hits a different circle skill. Show working. 3 marks each
1.1 For $x^2 + y^2 = 144$, state (a) the centre, (b) the radius, (c) the four axis crossings.
1.2 Write the equation of a circle centred at the origin that passes through (a) $(3, 4)$ (b) $(8, 6)$ (c) $(\sqrt{7}, 3)$.
1.3 Sort each equation into ONE of: "circle, $r = $ ?", "parabola", "no real graph", or "other (not a circle through origin)": (a) $x^2 + y^2 = 64$ (b) $y = x^2$ (c) $x^2 + y^2 = -25$ (d) $x^2 - y^2 = 9$ (e) $x^2 + y^2 = 1$.
1.4 The circle $x^2 + y^2 = 169$ passes through several Pythagorean-triple points. (a) Confirm $(5, 12)$ lies on it. (b) Confirm $(12, 5)$ lies on it. (c) Find another point $(a, b)$ with $a, b$ positive integers that also lies on this circle.
1.5 Rearrange $x^2 + y^2 = 25$ to make $y$ the subject. (a) Show $y = \pm\sqrt{25 - x^2}$. (b) Explain why the $+$ root only gives the UPPER half of the circle. (c) State the values of $x$ for which $y$ is real.
1.6 Two concentric circles (same centre, different radii) are centred at the origin: $x^2 + y^2 = 16$ and $x^2 + y^2 = 64$. (a) State the radius of each. (b) Where does the point $(0, 6)$ sit — inside both, between them, or outside both? Justify.
2. Find the mistake
A student has tried to write the equation of a circle centred at the origin passing through $(3, 4)$ and to decide whether $(4, 3)$ also lies on it. There are two mistakes — one in finding $r$, one in the on/off decision. Spot both, explain, and give the correct results. 3 marks
Student's working:
Step 1: $r = 3 + 4 = 7$.
Step 2: Equation: $x^2 + y^2 = 49$.
Step 3: Test $(4, 3)$: $4 + 3 = 7$, so on the circle ✓
(a) What is the correct value of $r$ for a circle through $(3, 4)$? Show your working.
(b) What is the correct equation of the circle?
(c) Apply the correct point-test to $(4, 3)$. Does it lie on the circle? Explain.
Stuck? You need $x^2 + y^2$, not $x + y$. So $r^2 = 3^2 + 4^2 = 25$, giving $r = 5$ (not $7$).3. Open-ended challenge — design a target board
This question has many valid answers. Be creative. 4 marks
3.1 You are designing a circular dartboard centred at the origin (units in cm) with FOUR concentric scoring rings: bullseye, inner, middle, outer.
(i) Choose four radii (in whole cm), smallest to largest, for the four boundary circles. Write the equation of EACH of the four circles.
(ii) Score: bullseye = 50 pts, inner ring = 25, middle = 10, outer = 5. State which equation corresponds to which boundary.
(iii) For each of the four dart positions below, compute $x^2 + y^2$ and decide which ring it lands in (or "off the board"): A $(0, 2)$, B $(4, 3)$, C $(7, 7)$, D $(12, 9)$.
Bonus: Your radii must be DIFFERENT from $1, 2, 3, 4$ (pick your own four).
How did this worksheet feel?
What I'll revisit before next class:
1.1 — $x^2 + y^2 = 144$
(a) Centre $(0, 0)$. (b) $r = \sqrt{144} = 12$. (c) Crossings: $(12, 0)$, $(-12, 0)$, $(0, 12)$, $(0, -12)$.
1.2 — Circles through given points
(a) $r^2 = 9 + 16 = 25$: $\mathbf{x^2 + y^2 = 25}$.
(b) $r^2 = 64 + 36 = 100$: $\mathbf{x^2 + y^2 = 100}$.
(c) $r^2 = 7 + 9 = 16$: $\mathbf{x^2 + y^2 = 16}$.
1.3 — Sort the equations
(a) Circle, $r = 8$. (b) Parabola. (c) No real graph (right side negative). (d) Other — the minus sign between $x^2$ and $y^2$ makes it a hyperbola, not a circle. (e) Circle, $r = 1$ (the "unit circle").
1.4 — Points on $x^2 + y^2 = 169$
(a) $5^2 + 12^2 = 25 + 144 = 169$ ✓
(b) $12^2 + 5^2 = 144 + 25 = 169$ ✓
(c) Several valid: $(0, 13)$ since $0 + 169 = 169$; or $(13, 0)$. (Other less-obvious positive-integer triple: none other than rearrangements of $(5, 12)$.) Accept any of $(0, 13)$, $(13, 0)$, $(5, 12)$ rearrangements.
1.5 — Make $y$ the subject of $x^2 + y^2 = 25$
(a) $y^2 = 25 - x^2 \Rightarrow y = \pm\sqrt{25 - x^2}$.
(b) The $+$ root gives only $y \ge 0$ values, so the graph sits on or above the $x$-axis — the UPPER semicircle.
(c) For $y$ to be real, need $25 - x^2 \ge 0$, i.e. $x^2 \le 25$, so $-5 \le x \le 5$.
1.6 — Two concentric circles
(a) $r_1 = 4$ (inner) and $r_2 = 8$ (outer).
(b) Point $(0, 6)$: $0^2 + 6^2 = 36$. Compare: $36 > 16$ (outside the inner) and $36 < 64$ (inside the outer). So it sits between the two circles — in the annular ring.
2 — Find the mistake
(a) For a circle centred at the origin through $(3, 4)$, $r^2 = 3^2 + 4^2 = 9 + 16 = 25$, so $r = 5$ (not $7$). The student added $3 + 4$ instead of $3^2 + 4^2$ — they used the wrong formula.
(b) Correct equation: $\mathbf{x^2 + y^2 = 25}$.
(c) Test $(4, 3)$: $4^2 + 3^2 = 16 + 9 = 25 = r^2$ — YES, $(4, 3)$ does lie on the circle. The student got the right answer for the wrong reason — their addition rule ($4 + 3 = 7$) happens to fail, but the proper $x^2 + y^2$ test confirms the point is on. (In general the $x + y$ rule fails — for instance $(5, 0)$ has $x + y = 5$ but is genuinely on the circle.)
3 — Open-ended challenge (sample solution)
Sample radii: bullseye $r = 3$, inner $r = 6$, middle $r = 10$, outer $r = 15$.
Equations: bullseye $x^2 + y^2 = 9$ (50 pts); inner $x^2 + y^2 = 36$ (25 pts); middle $x^2 + y^2 = 100$ (10 pts); outer $x^2 + y^2 = 225$ (5 pts).
Dart placement: A $(0, 2)$: $0 + 4 = 4 \le 9 \Rightarrow$ bullseye, 50 pts. B $(4, 3)$: $16 + 9 = 25$, $25 > 9$ but $25 \le 36 \Rightarrow$ inner, 25 pts. C $(7, 7)$: $49 + 49 = 98$, $98 > 36$ but $98 \le 100 \Rightarrow$ middle, 10 pts. D $(12, 9)$: $144 + 81 = 225 \le 225 \Rightarrow$ on the outer rim, 5 pts.
Marking: 1 mark per valid set of four equations, 1 mark for correct point classifications, 1 mark for working shown, 1 mark for use of different radii from the bonus restriction.