question_answer

If OA and OB be the tangents to the circle \[{{x}^{2}}+{{y}^{2}}-6x-8y+21=0\]drawn from the origin O,  then AB =

A)            11  

B)            \[\frac{4}{5}\sqrt{21}\]

C)            \[\sqrt{\frac{17}{3}}\]            

D)            None of these

Correct Answer: B

Solution :

           Here the equation of AB (chord of contact) is                    \[0+0-3(x+0)-4(y+0)+21=0\]                    \[\Rightarrow 3x+4y-21=0\]                                     ?. (i)                    CM = perpendicular distance from (3, 4) to line (i) is                    \[\frac{3\times 3+4\times 4-21}{\sqrt{9+16}}=\frac{4}{5}\]                    \[AM=\sqrt{A{{C}^{2}}-C{{M}^{2}}}=\sqrt{4-\frac{16}{25}}=\frac{2}{5}\sqrt{21}\]                    \[\therefore \ \ AB=2AM=\frac{4}{5}\sqrt{21}\] .