Certainly! Let's find the equation of the chord step-by-step.
### Step 1: Convert the equation of the circle to standard form
The given circle equation is:
[tex]\[ x^2 + y^2 - 6x - 2y - 15 = 0 \][/tex]
We need to rearrange it into the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex].
1. Group the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms:
[tex]\[ x^2 - 6x + y^2 - 2y = 15 \][/tex]
2. Complete the square for both [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
- For [tex]\(x^2 - 6x\)[/tex]:
[tex]\[ x^2 - 6x = (x - 3)^2 - 9 \][/tex]
- For [tex]\(y^2 - 2y\)[/tex]:
[tex]\[ y^2 - 2y = (y - 1)^2 - 1 \][/tex]
3. Substitute back:
[tex]\[ (x - 3)^2 - 9 + (y - 1)^2 - 1 = 15 \][/tex]
4. Simplify:
[tex]\[ (x - 3)^2 + (y - 1)^2 = 25 \][/tex]
So, the center of the circle [tex]\(C\)[/tex] is [tex]\((3, 1)\)[/tex] and the radius is 5.
### Step 2: Find the slope of the chord
The midpoint [tex]\(M\)[/tex] of the chord is given as [tex]\(M(2, 3)\)[/tex].
The slope of the line connecting the center [tex]\(C(3, 1)\)[/tex] to the midpoint [tex]\(M(2, 3)\)[/tex] is:
[tex]\[ \text{slope} = \frac{3 - 1}{2 - 3} = \frac{2}{-1} = -2 \][/tex]
### Step 3: Write the equation of the chord using the point-slope form
The point-slope form of a line equation is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, [tex]\(m\)[/tex] is the slope and [tex]\((x_1, y_1)\)[/tex] is the midpoint [tex]\(M(2, 3)\)[/tex]. We substitute the known values:
[tex]\[ y - 3 = -2(x - 2) \][/tex]
### Step 4: Rearrange to get the slope-intercept form
[tex]\[ y - 3 = -2x + 4 \][/tex]
[tex]\[ y = -2x + 4 + 3 \][/tex]
[tex]\[ y = -2x + 7 \][/tex]
### Conclusion
The equation of the chord is:
[tex]\[ y = -2x + 7 \][/tex]
That is the required linear equation representing the chord of the circle passing through the midpoint [tex]\(M(2, 3)\)[/tex].
