question_answer

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

A) \[\frac{7}{2}\]

B) \[\frac{3\sqrt{3}}{2}\]

C) \[\sqrt{14}\]

D) \[\frac{\sqrt{7}}{2}\]

Correct Answer: C

Solution :

[c] The centre of the circle is \[C(1,1)\] and radius of the circle is 2, perpendicular distance from C on AB, the chord\[x+y=3\] \[CD=\left| \frac{1+1-3}{\sqrt{2}} \right|=\frac{1}{\sqrt{2}}\] \[\therefore AD=\sqrt{4-\frac{1}{2}}=\sqrt{\frac{7}{2}}\]\[[AD=\sqrt{A{{C}^{2}}-C{{D}^{2}}}]\] Hence, the length of the chord \[AB=2AD=2\sqrt{\frac{7}{2}}=\sqrt{14}\]