Question

The co-ordinates 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}{2}\right) $
• C. $\left(\frac{1}{2},\frac{1}{3}\right)$
• D. $(-1,0)$

Answer 1

Answer: (A)

The smallest circle means that its radial is, 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 $(0,0)$.
$\therefore \,\,\,\,\,\lambda=0$
$\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.