Which of the following is the equation of a circle in the standard (x,y) coordinate plane ?

Options

A. (x-3)+(y+6)=100
B. `y=x^2+25`
C. `x^2=y^2+25`
D. `x^2=36-y^2`

Answer

To determine which of the given options is the equation of a circle in the standard (x,y) coordinate plane, we need to recall the general equation of a circle. The standard form of the equation of a circle is:

\[ (x - h)^2 + (y - k)^2 = r^2 \]

where \((h, k)\) is the center of the circle and \(r\) is the radius. If the center is at the origin (0, 0), the equation simplifies to:

\[ x^2 + y^2 = r^2 \]

Now, let's analyze each option provided in the question.

### Step 1: Analyze Option A
- **Equation:** \(y = 5\)
- **Analysis:** This equation does not contain any squared terms. Therefore, it cannot represent a circle.

**Conclusion:** Option A is not the equation of a circle.

### Step 2: Analyze Option B
- **Equation:** \(x = 4\)
- **Analysis:** Similar to option A, this equation also does not contain any squared terms. Hence, it cannot represent a circle.

**Conclusion:** Option B is not the equation of a circle.

### Step 3: Analyze Option C
- **Equation:** \(x^2 = y^2 + 25\)
- **Rearranging:** Rearranging gives us \(x^2 - y^2 = 25\).
- **Analysis:** This is a difference of squares and does not match the form of a circle's equation. The terms are not both squared and added together.

**Conclusion:** Option C is not the equation of a circle.

### Step 4: Analyze Option D
- **Equation:** \(x^2 = 36 - y^2\)
- **Rearranging:** Rearranging gives us \(x^2 + y^2 = 36\).
- **Analysis:** This matches the standard form of a circle's equation where \(h = 0\), \(k = 0\), and \(r^2 = 36\). Therefore, the radius \(r = 6\).

**Conclusion:** Option D is the equation of a circle.

### Final Answer
The equation of a circle in the standard (x,y) coordinate plane is given by **Option D: \(x^2 + y^2 = 36\)**.

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