The length of the chord $x + y = 3$ intercepted by the circle $x^{2} + y^{2} - 2 x - 2 y - 2 = 0$ is

\frac{7}{2}

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\frac{3 \sqrt{3}}{2}

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\sqrt{14}

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\frac{\sqrt{7}}{2}

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Solution

The correct option is D $\sqrt{14}$
The centre of the circle is $C (1, 1)$ and radius of the circle is $2$ , perpendicular distance from $C$ on $A B$ , the chord $x + y = 3$
$C D = ∣ ∣ ∣ \frac{1 + 1 - 3}{\sqrt{2}} ∣ ∣ ∣ = \frac{1}{\sqrt{2}}$
$\Rightarrow A D = \sqrt{4 - \frac{1}{2}} = \sqrt{\frac{7}{2}} [A D = \sqrt{A C^{2} - C D^{2}}]$
Hence, the length of the chord $A B = 2 A D = 2 \sqrt{\frac{7}{2}} = \sqrt{14}$ .

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