Question

If a circle passes through the points $(0,0)$, $(x, 0)$ and $(0, y)$, then the coordinates of its centre are

• A. $\left(-\frac{x}{2}, \frac{y}{2}\right)$
• B. $\left(\frac{x}{2}, \frac{y}{2}\right)$
• C. $\left(-\frac{x}{2}, -\frac{y}{2}\right)$
• D. $\left(\frac{x}{2}, -\frac{y}{2}\right)$

Hint:

Whenever a circle passes through the origin and cuts intercepts of lengths $a$ and $b$ on the coordinate axes, the endpoints of its diameter will always be $(a,0)$ and $(0,b)$. Consequently, its center will always simplify directly to $(\frac{a}{2}, \frac{b}{2})$.

Answer 1

The Correct Option is B

Step 1: Set up the general circle.
Let the circle be $x^2 + y^2 + 2gx + 2fy + c = 0$, where the centre is $(-g, -f)$. We will use the three given points to find $g$, $f$ and $c$.
Step 2: Use the origin.
Substituting $(0,0)$ gives $0 + 0 + 0 + 0 + c = 0$, so $c = 0$. The circle passes through the origin, so the constant term vanishes.
Step 3: Use the point $(x, 0)$.
Substituting gives $x^2 + 0 + 2gx + 0 + 0 = 0$. Since $x \neq 0$, divide by $x$ to get $x + 2g = 0$, hence $g = -\dfrac{x}{2}$.
Step 4: Use the point $(0, y)$.
Substituting gives $0 + y^2 + 0 + 2fy + 0 = 0$. Since $y \neq 0$, divide by $y$ to get $y + 2f = 0$, hence $f = -\dfrac{y}{2}$.
Step 5: Read off the centre.
The centre is $(-g, -f) = \left(\dfrac{x}{2}, \dfrac{y}{2}\right)$.
Step 6: Quick check.
Because the two axes meet at a right angle at the origin, the chord joining $(x,0)$ and $(0,y)$ is a diameter, and its midpoint $\left(\dfrac{x}{2}, \dfrac{y}{2}\right)$ is indeed the centre. Both methods agree.
\[ \boxed{\left(\dfrac{x}{2}, \dfrac{y}{2}\right)} \]