The circle `x^2+y^2−8x+4y+4=0` touches: a) x axis b) y axis c) both the axis d) neither x axis nor y axis

Answer

To solve the given problem, we need to analyze the equation of the circle and determine its center and radius. The equation of the circle is given as:

\[ x^2 + y^2 - 8x + 4y + 4 = 0 \]

### Step 1: Rearranging the Circle's Equation
We will rearrange the equation to identify the center and radius of the circle.

First, we can rewrite the equation as:

\[ x^2 - 8x + y^2 + 4y + 4 = 0 \]

### Step 2: Completing the Square
Next, we will complete the square for the \(x\) and \(y\) terms.

1. For \(x^2 - 8x\):
- Take half of \(-8\), which is \(-4\), and square it to get \(16\).
- Thus, \(x^2 - 8x\) can be rewritten as \((x - 4)^2 - 16\).

2. For \(y^2 + 4y\):
- Take half of \(4\), which is \(2\), and square it to get \(4\).
- Thus, \(y^2 + 4y\) can be rewritten as \((y + 2)^2 - 4\).

Now substituting these back into the equation, we have:

\[ (x - 4)^2 - 16 + (y + 2)^2 - 4 + 4 = 0 \]

This simplifies to:

\[ (x - 4)^2 + (y + 2)^2 - 16 = 0 \]

Rearranging gives:

\[ (x - 4)^2 + (y + 2)^2 = 16 \]

### Step 3: Identifying the Center and Radius
From the equation \((x - 4)^2 + (y + 2)^2 = 4^2\), we can identify:

- The center of the circle \((h, k)\) is \((4, -2)\).
- The radius \(r\) is \(4\).

### Step 4: Analyzing Touching Conditions
Now, we need to determine whether the circle touches the x-axis, y-axis, both, or neither.

1. **Touching the x-axis**: The distance from the center to the x-axis is the absolute value of the y-coordinate of the center, which is \(|-2| = 2\). Since the radius is \(4\), the circle does touch the x-axis.

2. **Touching the y-axis**: The distance from the center to the y-axis is the absolute value of the x-coordinate of the center, which is \(|4| = 4\). Since the radius is also \(4\), the circle touches the y-axis as well.

### Conclusion
Since the circle touches both the x-axis and the y-axis, the correct answer is:

**c) both the axes**