question_answer

The coordinates of the centre of the smallest circle passing through the origin and having \[y=x+1\] as a diameter are

A)  \[\left( \frac{1}{2},\frac{-1}{2} \right)\]                

B)  \[\left( \frac{1}{2},\frac{1}{3} \right)\]

C)  \[(-1,0)\]                            

D)  \[\left( -\frac{1}{2},\frac{1}{2} \right)\]

Correct Answer: D

Solution :

The smallest circle means that is radial is, distance from origin to the diameter id smaller.   Let equation of line perpendicular to \[y-x=1\] is \[x+y=\lambda \] Also, it passes through \[(0,0)\]. \[\therefore \]  \[\lambda =0\] \[\therefore \] Perpendicular line is \[x+y=0.\] The intersection point of lines are \[\left( -\frac{1}{2},\frac{1}{2} \right)\] Which is the centre of circle.