Question

The centre of a circle is ($2a,a-7)$. Find the values of a if the circle passes through the point (11,$-$9) and has diameter $10\sqrt{2}$ units.

Answer 1

The length of the diameter is $10\sqrt{2}units$.
So, the radius is $5\sqrt{2}units$.
The centre of the circle be C(2a, a−7).
Suppose it passes through the point P(11, −9).
Therefore, PC = r
$$\begin{gathered} \Rightarrow P{C}^2=r^2 \\ \Rightarrow {\left(11-2a\right)}^2+{\left(-9-a+7\right)}^2={\left(5\sqrt{2}\right)}^2 \\ \Rightarrow 121+4a^2-44a+a^2+4+4a=50 \\ \Rightarrow 5a^2-40a+75=0 \\ \Rightarrow \left(a-3\right)\left(a-5\right)=0 \\ \Rightarrow a=3ora=5 \end{gathered}$$
Hence the values of a are 3 or 5.