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A circle centered at $(2, 1)$ passes through the points A$(5, 6)$ and B($-3$, K). Find the value(s) of K. Hence find length of chord AB.
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Sol. Let centre be O$(2,1) \Rightarrow OA = OB$ (1 Mark)
$\sqrt{(5 - 2)^2 + (6 - 1)^2} = \sqrt{(-3-2)^2 + (K - 1)^2}$ (1 Mark)
$\Rightarrow 9 = (K - 1)^2$
$\Rightarrow K = -2,4$ (1/2 Mark for each value of K)
For K = $-2$, AB = $\sqrt{128}$ or $8\sqrt{2}$ (1/2 Mark)
For K = $4$, AB = $\sqrt{68}$ or $2\sqrt{17}$ (1/2 Mark)
$\sqrt{(5 - 2)^2 + (6 - 1)^2} = \sqrt{(-3-2)^2 + (K - 1)^2}$ (1 Mark)
$\Rightarrow 9 = (K - 1)^2$
$\Rightarrow K = -2,4$ (1/2 Mark for each value of K)
For K = $-2$, AB = $\sqrt{128}$ or $8\sqrt{2}$ (1/2 Mark)
For K = $4$, AB = $\sqrt{68}$ or $2\sqrt{17}$ (1/2 Mark)