KCET · Maths · Circle
The coordinates of the centre of the smallest circle passing through the origin and having \(\mathrm{y}=\mathrm{x}+1\) 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 \(\left(-\frac{1}{2}, \frac{1}{2}\right)\)
Answer & Solution
Correct Answer
(D) \(\left(-\frac{1}{2}, \frac{1}{2}\right)\)
Step-by-step Solution
Detailed explanation
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 \(\mathrm{x}+\mathrm{y}=\lambda\).
Also, it passes through \((0,0)\).
\(\therefore \quad \lambda=0\)
\(\therefore\) Perpendicular line is \(\mathrm{x}+\mathrm{y}=0\).
The intersection point of lines are \(\left(-\frac{1}{2}, \frac{1}{2}\right)\).
Which is the centre of circle.
Let equation of line perpendicular to \(y-x=1\) is \(\mathrm{x}+\mathrm{y}=\lambda\).
Also, it passes through \((0,0)\).
\(\therefore \quad \lambda=0\)
\(\therefore\) Perpendicular line is \(\mathrm{x}+\mathrm{y}=0\).
The intersection point of lines are \(\left(-\frac{1}{2}, \frac{1}{2}\right)\).
Which is the centre of circle.
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