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

Options

A. `((1)/(2),-(1)/2)`
B. `((1)/(2),-(1)/3)`
C. (-1,0)
D. `(-(1)/(2),-(1)/2)`

Answer

D

Explanation

The smallest circle means that, its radial distance from origin to the diameter is smallest.
let equation of line perpendicular to y-x=1 is x +y =`lambda.` also, it passes through (0,0). `
:. lambda =0`
so, perpendicular line is x +y=0. the intersection point of line are `(-1/2,1/2),` which is the centreof circle. Alternate method
it is clear from the figure that, centrwe lies on llnd quadrant. hence, option (d) is correct.