CAIE IGCSE Additional Math 8 Coordinate Geometry of the Circle Topic Practice

$$\begin{aligned} (x-2)^{2}+(y+1)^{2}=74 \text { oe nfww } \text { or } x^{2}+y^{2}-4 x+2 y-69=0 \text { oe nfww } \end{aligned}$$

M1 for centre: $\left(\frac{-3+7}{2}, \frac{6-8}{2}\right)$ or (2,-1) soi

M2 for $[r=] \sqrt{74}$ or $\left[r^{2}=\right] 74$ or [diameter =] $2 \sqrt{74}$ or c=-69

or M1 for $\left[A B^{2}=\right](-3-7)^{2}+(6--8)^{2}$ oe or $\left[r^{2}=\right](-3-\text { their } 2)^{2}+(6-\text { their }(-1))^{2}$ oe or $\left[r^{2}=\right](7-\text { their } 2)^{2}+(-8-\text { their }(-1))^{2}$ oe

Alternative method

M2 for $\frac{y-6}{x-(-3)} \times \frac{y-(-8)}{x-7}=-1$

or M1 for e.g. P(x, y) on the circumference means $A P \perp B P \rightarrow m_{A P} \times m_{B P}=-1$

M1 for $\frac{y^{2}+2 y-48}{x^{2}-4 x-21}=-1$ oe

FT their $\left(\frac{y-6}{x-(-3)} \times \frac{y-(-8)}{x-7}\right)$ with at most one sign error

Radius: 5 soi
Centre: $(4\sqrt2,4\sqrt2)$ soi oe
B1 for (5.66,5.66) or better, or for distance between centres is 8 soi.
$$\left(x-\frac{8}{\sqrt2}\right)^2+\left(y-\frac{8}{\sqrt2}\right)^2=25$$
or equivalent expansion.
Final answer: $x^2+y^2-8\sqrt2x-8\sqrt2y+39=0$
Must be exact.
B1, B2, M1, A1 with FT alternatives as in markscheme.