question_answer

The line segment joining the points (4, 7) and \[(-2,-1)\]is a diameter of a circle. If the circle intersects the x-axis at A and B, then AB is equal to

A)  4                                            

B)  5

C)  6                            

D)         7

E)  8

Correct Answer: E

Solution :

\[\because \] (4, 7) and\[(-2,-1)\]are end points of circle. \[\Rightarrow \]Centre of circle                 \[=\left( \frac{4-2}{2},\frac{7-1}{2} \right)=(1,3)\] \[\therefore \]Slope of\[AC=\frac{-1-0}{-2-a}=-\frac{1}{2+a}\] and slope of\[AD=\frac{7-0}{4-a}=\frac{7}{4-a}\] Since,\[\Delta CAD\]is right angle as CD is diameter. \[\Rightarrow \](slope of AC (slope of AD)\[=-1\] \[\Rightarrow \]               \[\left( \frac{1}{2+a} \right)\left( \frac{7}{4-a} \right)=-1\] \[\Rightarrow \]               \[\frac{7}{-{{a}^{2}}+2a+8}=-1\] \[\Rightarrow \]               \[{{a}^{2}}-2a-15=0\] \[\Rightarrow \]               \[a=-3,5\] Similarly, \[b=-3,5\] \[\therefore \]Point\[A=(-3,0)\]and point\[B=(5,0)\] \[\therefore \]  \[AB=\sqrt{{{(-3-5)}^{2}}+{{(0-0)}^{2}}}\]                 \[=\sqrt{{{(-8)}^{2}}}=8\]