Question

# The length of the common chord of the two circles $$x^2+y^2-4y=0$$ and $$x^2+y^2-8x-4y+11=0$$ is

A
1454
B
112
C
135
D
1354

Solution

## The correct option is D $$\dfrac{\sqrt{135}}{4}$$$$s_1=x^2+y^2-4y=0$$$$s_2=x^2+y^2-8x-4y+11=0$$Equation common chord $$=s_1-s_2=8x-11$$$$OC= \left|\dfrac{-11}{\sqrt{8^2}}\right|=\dfrac{11}{8}$$$$CD=\left|\dfrac{32-11}{\sqrt{8^2}}\right|=\left|\dfrac{21}{8}\right|$$$$AC^2=OA^2-OC^2$$$$=2^2-\left(\dfrac{11}{8}\right)^2$$$$=\dfrac{256-121}{64}$$$$=\dfrac{135}{64}$$$$AC=\sqrt{\dfrac{135}{64}}=\sqrt{\dfrac{135}{8}}$$$$2AC=\dfrac{\sqrt{135}}{4}$$ is the length of common chord.Mathematics

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