Find the equation of the chord of `x^(2)+y^(2)-6x+10y-9=0` which is bisected at (-2,4)

Answer

To find the equation of the chord of the circle given by \(x^2 + y^2 - 6x + 10y - 9 = 0\) that is bisected at the point \((-2, 4)\), we can follow these steps:

### Step 1: Rewrite the Circle's Equation
First, we need to rewrite the equation of the circle in standard form. The given equation is:
\[ x^2 + y^2 - 6x + 10y - 9 = 0 \]
We can rearrange it to group the \(x\) and \(y\) terms:
\[ x^2 - 6x + y^2 + 10y = 9 \]

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

For \(x\):
\[ x^2 - 6x \rightarrow (x - 3)^2 - 9 \]

For \(y\):
\[ y^2 + 10y \rightarrow (y + 5)^2 - 25 \]

Substituting these back into the equation gives:
\[ (x - 3)^2 - 9 + (y + 5)^2 - 25 = 9 \]
Simplifying this, we have:
\[ (x - 3)^2 + (y + 5)^2 - 34 = 9 \]
\[ (x - 3)^2 + (y + 5)^2 = 43 \]
This shows that the circle has center \((3, -5)\) and radius \(\sqrt{43}\).

### Step 3: Find the Slope of the Radius
The radius from the center \((3, -5)\) to the midpoint of the chord \((-2, 4)\) can be used to find the slope of the radius. The slope \(m\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - (-5)}{-2 - 3} = \frac{9}{-5} = -\frac{9}{5} \]

### Step 4: Find the Slope of the Chord
The slope of the chord, which is perpendicular to the radius, is the negative reciprocal of the slope of the radius:
\[ m' = -\frac{1}{m} = -\frac{1}{-\frac{9}{5}} = \frac{5}{9} \]

### Step 5: Use the Point-Slope Form to Write the Equation of the Chord
Using the point-slope form of the equation of a line, we can write the equation of the chord that passes through the midpoint \((-2, 4)\):
\[ y - y_1 = m'(x - x_1) \]
Substituting the values:
\[ y - 4 = \frac{5}{9}(x + 2) \]

### Step 6: Simplify the Equation
Now, we simplify the equation:
\[ y - 4 = \frac{5}{9}x + \frac{10}{9} \]
Multiplying through by 9 to eliminate the fraction:
\[ 9y - 36 = 5x + 10 \]
Rearranging gives:
\[ 5x - 9y + 46 = 0 \]

### Final Equation
Thus, the equation of the chord is:
\[ 5x - 9y + 46 = 0 \]

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