Which of the following are the coordinates of the left - most point on the circle represented by the equations `x^(2) + y^(2) - 8x + 6y = 0 ` ?

Options

A. `(9-3)`
B. `(1,3)`
C. `(-1,-3)`
D. `(-9,-3)`

Answer

To find the coordinates of the left-most point on the circle represented by the equation \( x^2 + y^2 - 8x + 6y = 0 \), we will follow these steps:

### Step 1: Rewrite the Circle Equation
We start by rewriting the given circle equation in standard form. The standard form of a circle's equation is:

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

where \((h, k)\) is the center of the circle and \(r\) is the radius.

### Step 2: Complete the Square
We need to complete the square for both \(x\) and \(y\) in the equation \(x^2 + y^2 - 8x + 6y = 0\).

1. For \(x\):
\[ x^2 - 8x \quad \text{can be rewritten as} \quad (x - 4)^2 - 16 \]

2. For \(y\):
\[ y^2 + 6y \quad \text{can be rewritten as} \quad (y + 3)^2 - 9 \]

Putting it all together, we have:
\[ (x - 4)^2 - 16 + (y + 3)^2 - 9 = 0 \]
This simplifies to:
\[ (x - 4)^2 + (y + 3)^2 = 25 \]

### Step 3: Identify the Center and Radius
From the standard form \((x - 4)^2 + (y + 3)^2 = 25\), we can identify:
- The center of the circle \((h, k) = (4, -3)\)
- The radius \(r = \sqrt{25} = 5\)

### Step 4: Find the Left-Most Point
The left-most point on the circle can be found by moving left from the center by the radius. The center is at \( (4, -3) \), and moving left by the radius of 5 gives us:

\[ \text{Left-most point} = (4 - 5, -3) = (-1, -3) \]

### Conclusion
The coordinates of the left-most point on the circle are \((-1, -3)\).

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