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: Understanding the Question:
The problem describes a circle that passes through the coordinate origin $(0,0)$, a point on the x-axis $(x, 0)$, and a point on the y-axis $(0, y)$. We need to find the coordinates of the center of this circle.

Step 2: Key Formula or Approach:
Consider the geometric configuration of the points. The point $(x,0)$ lies on the x-axis, and the point $(0,y)$ lies on the y-axis. The lines connecting the origin to these points form a right angle ($90^\circ$) at the origin $(0,0)$. According to Thales's Theorem, an angle inscribed in a semicircle is a right angle. Therefore, the line segment connecting $(x,0)$ and $(0,y)$ must be a diameter of the circle. The center of the circle is simply the midpoint of this diameter segment.

Step 3: Detailed Explanation:
Let the three given points on the circle circumference be: $O(0, 0)$ $P(x, 0)$ $Q(0, y)$ Since the x-axis and y-axis intersect at a perfect right angle ($90^\circ$), $\angle POQ = 90^\circ$. By the chord-angle properties of circles, any chord that subtends a $90^\circ$ angle on the circumference must be a diameter of that circle. Thus, the line segment $PQ$ acts as a diameter. The coordinates of the center of a circle are the midpoint of its diameter. Apply the standard midpoint formula: $$\text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$ Substitute the coordinates of points $P(x,0)$ and $Q(0,y)$: $$\text{Center} = \left(\frac{x + 0}{2}, \frac{0 + y}{2}\right) = \left(\frac{x}{2}, \frac{y}{2}\right)$$

Step 4: Final Answer:
The coordinates of the center of the circle are $\left(\frac{x}{2}, \frac{y}{2}\right)$, which corresponds to option (B).