The set of circles passing through (0,0) is :

Options

A. Infinite Set
B. Finite set
C. Null set
D. None of these.

Answer

To determine the set of circles passing through the point (0,0), we can follow these steps:

### Step 1: Understand the general equation of a circle
The general equation of a circle with center (h, k) and radius r is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]

### Step 2: Set the center at the origin
Since we are interested in circles that pass through the origin (0,0), we set the center of the circle at the origin. Therefore, h = 0 and k = 0. The equation simplifies to:
\[ x^2 + y^2 = r^2 \]

### Step 3: Identify the radius
The radius r can be any positive real number. This means that r can take any value from 0 to infinity (r > 0).

### Step 4: Formulate the set of circles
Thus, the set of circles passing through the origin can be described as:
\[ \{ (x, y) : x^2 + y^2 = r^2, r > 0 \} \]
This indicates that for every positive radius r, there exists a circle.

### Step 5: Conclusion about the set
Since r can take infinitely many values, the set of circles passing through (0,0) is infinite. Therefore, we conclude that the set of circles passing through the origin is an infinite set.

### Final Answer
The set of circles passing through (0,0) is an infinite set of the form:
\[ \{ x^2 + y^2 = r^2 \, | \, r > 0 \} \]

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