Question

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

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Solution

## 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 $⇒P{C}^{2}={r}^{2}\phantom{\rule{0ex}{0ex}}⇒{\left(11-2a\right)}^{2}+{\left(-9-a+7\right)}^{2}={\left(5\sqrt{2}\right)}^{2}\phantom{\rule{0ex}{0ex}}⇒121+4{a}^{2}-44a+{a}^{2}+4+4a=50\phantom{\rule{0ex}{0ex}}⇒5{a}^{2}-40a+75=0\phantom{\rule{0ex}{0ex}}⇒\left(a-3\right)\left(a-5\right)=0\phantom{\rule{0ex}{0ex}}⇒a=3ora=5$ Hence the values of a are 3 or 5.

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