A-Level Mathematics October/November 2024 Q6(a): Circles C₁ and C₂ have equations x² + y² + 6x-10y+18 = 0 and (x-9)² + (y + 4)² – 64 = 0…

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  First, find the centre of circle $C_1$ by completing the square. $x^2 + y^2 + 6x - 10y + 18 = 0$ $(x^2 + 6x) + (y^2 - 10y) + 18 = 0$ $((x+3)^2 - 3^2) + ((y-5)^2 - 5^2) + 18 = 0$ $(x+3)^2 - 9 + (y-5)^2 - 25 + 18 = 0$ $(x+3)^2 + (y-5)^2 = 16$ The centre of $C_1$ is $(-3, 5)$.

  Next, find the centre of circle $C_2$ by rearranging its equation into standard form. $(x-9)^2 + (y + 4)^2 - 64 = 0$ $(x-9)^2 + (y+4)^2 = 64$ The centre of $C_2$ is $(9, -4)$.

  Finally, find the distance between the two centres, $(-3, 5)$ and $(9, -4)$, using the distance formula. Distance $= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ Distance $= \sqrt{(9 - (-3))^2 + (-4 - 5)^2}$ Distance $= \sqrt{(12)^2 + (-9)^2}$ Distance $= \sqrt{144 + 81}$ Distance $= \sqrt{225}$ Distance $= 15$

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  How the marks are awarded

  • B1 — Awarded for correctly identifying the centre of circle $C_1$ as $(-3, 5)$, typically by completing the square.
  • B1 — Awarded for correctly identifying the centre of circle $C_2$ as $(9, -4)$ from its equation.
  • M1 — Awarded for applying the correct distance formula, $\sqrt{(\Delta x)^2 + (\Delta y)^2}$, to the two centres found by the student.
  • A1 — Awarded for obtaining the correct final answer of 15 from correct working.

  Common mistakes

  • Sign errors when finding the centre from the equation. For $C_1$, stating the centre is $(3, -5)$ instead of $(-3, 5)$. For $C_2$, stating $(-9, 4)$ instead of $(9, -4)$.
  • Incorrectly applying the distance formula, for example by not squaring the differences inside the square root, or mixing up x and y coordinates.
  • Arithmetic errors when calculating the final distance, such as $12^2 + (-9)^2 = 144 - 81$ or incorrectly finding the square root of 225.
  • Confusing the general form $x^2+y^2+2gx+2fy+c=0$ where the centre is $(-g, -f)$, and misreading $2g=6$ as $g=6$, leading to an incorrect centre.

  Examiner tip: Ensure you can confidently convert a circle's equation from general form to standard form $(x-a)^2 + (y-b)^2 = r^2$ to quickly and accurately identify its centre and radius.

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