For the circle `x ^(2) + y^(2) + 6x -8y +9=0,` which of the following statement is true ?

Options

A. Circle passes through the point `(-3,4)`
B. Circle touches X-axis
C. Circle touches Y-axis
D. None of these

Answer

To solve the problem, we need to analyze the given circle equation:

**Step 1: Rewrite the equation of the circle in standard form.**

The given equation is:
\[ x^2 + y^2 + 6x - 8y + 9 = 0 \]

We can rearrange it to group the \(x\) and \(y\) terms:
\[ x^2 + 6x + y^2 - 8y + 9 = 0 \]

**Step 2: Complete the square for the \(x\) terms.**

For the \(x\) terms \(x^2 + 6x\):
- Take half of the coefficient of \(x\) (which is 6), square it, and add/subtract it:
\[ x^2 + 6x = (x + 3)^2 - 9 \]

**Step 3: Complete the square for the \(y\) terms.**

For the \(y\) terms \(y^2 - 8y\):
- Take half of the coefficient of \(y\) (which is -8), square it, and add/subtract it:
\[ y^2 - 8y = (y - 4)^2 - 16 \]

**Step 4: Substitute back into the equation.**

Now substitute the completed squares back into the equation:
\[ (x + 3)^2 - 9 + (y - 4)^2 - 16 + 9 = 0 \]

Combine the constants:
\[ (x + 3)^2 + (y - 4)^2 - 16 = 0 \]
\[ (x + 3)^2 + (y - 4)^2 = 16 \]

**Step 5: Identify the center and radius of the circle.**

From the standard form \((x - h)^2 + (y - k)^2 = r^2\), we can see:
- Center \((h, k) = (-3, 4)\)
- Radius \(r = \sqrt{16} = 4\)

**Step 6: Analyze the position of the circle relative to the axes.**

- The center of the circle is at \((-3, 4)\).
- The radius is \(4\), which means the circle extends from:
- Left: \(-3 - 4 = -7\) (touching the y-axis)
- Right: \(-3 + 4 = 1\)
- Up: \(4 + 4 = 8\)
- Down: \(4 - 4 = 0\) (touching the x-axis)

**Step 7: Determine the statements.**

- The circle touches the x-axis at the point \((-3, 0)\) because the y-coordinate of the center (4) is equal to the radius (4).
- The circle intersects the y-axis at the point \((0, 4)\) because the distance from the center to the y-axis is \(3\) (which is less than the radius).

Based on this analysis, the true statement is:
- The circle touches the x-axis.