A and B are two points in xy-plane which are $2 \sqrt{2}$ unit distance apart and subtended an angle of 90⁰ at the point C(1, 2) on the line $x - y + 1 = 0$ which is larger than any angle subtended by the line segment AB at any other point on the line. The equation of the circle through the points A, B and C is ...........

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$x^{2} + y^{2} - 6 y + 7 = 0$

$x^{2} + y^{2} - 2 x - y + 3 = 0$

$x^{2} + y^{2} - 6 x + 6 y + 7 = 0$

$x^{2} + y^{2} - 4 x - 2 y + 3 = 0$

answer is A.

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Detailed Solution

$\bar{AB}$ subtends the greatest angle at C on the line $x - y + 1 = 0$ .

Therefore $x - y + 1 = 0$ touches the circle at C and hence $\bar{AB}$ as diameter .

required circle be $(x - 1)^{2} + (y - 2)^{2} + λ (x - y + 1) = 0$

$x^{2} + y^{2} + x (λ - 2) + y (- λ - 4) + (λ + 5) = 0$

$A B = 2 \sqrt{2} \Rightarrow radius = \sqrt{2}$

$\frac{{(λ - 2)}^{2}}{4} + \frac{{(λ + 4)}^{2}}{4} - (λ + 5) = 2 \Rightarrow 2 λ^{2} + 4 λ + 20 - 4 λ - 20 = 8 \Rightarrow λ^{2} = 4$

$\Rightarrow λ = \pm 2$

Equation of circle is $x^{2} + y^{2} - 6 y + 7 = 0$

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