The centre of a circle is $(3 k + 1, 2 k - 1)$ . If the circle passes through the point $(- 1, - 3)$ and the length of its diameter be $20$ units, find the value of k.

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Solution

Centre $\Rightarrow C (3 k + 1, 2 k - 1), P = (- 1, - 3)$
$P C =$ radius $= 10$
$\Rightarrow {P C}^{2} = 100$
$\Rightarrow {[- 3 - (2 k - 1)]}^{2} + {[- 1 - (3 k + 1)]}^{2} = 100$
$\Rightarrow {(2 k + 2)}^{2} + {(3 k - 2)}^{2} = 100$
$\Rightarrow {4 k}^{2} + 4 + 8 k + {9 k}^{2} + 4 - 12 k = 100$
$\Rightarrow {13 k}^{2} - 4 k + 8 = 100$
$\Rightarrow {13 k}^{2} - 4 k - 92 = 0$
$\Rightarrow k = \frac{2}{13} \pm \frac{20}{13} \sqrt{3}$

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