The distance of the point (1,-2) from the common chord of the circles
` x^(2) + y^(2) - 5x +4y - 2= 0 " and " x^(2) +y^(2) - 2x + 8y + 3 = 0 `

Options

A. 2
B. 4
C. `1//2`
D. 0

Answer

To find the distance of the point (1, -2) from the common chord of the two given circles, we will follow these steps:

### Step 1: Write the equations of the circles
The equations of the circles are given as:
1. \( S_1: x^2 + y^2 - 5x + 4y - 2 = 0 \)
2. \( S_2: x^2 + y^2 - 2x + 8y + 3 = 0 \)

### Step 2: Find the equation of the common chord
The equation of the common chord can be found by subtracting the two circle equations:
\[ S_1 - S_2 = 0 \]
Substituting the equations:
\[ (x^2 + y^2 - 5x + 4y - 2) - (x^2 + y^2 - 2x + 8y + 3) = 0 \]
Simplifying this:
\[ -5x + 4y - 2 + 2x - 8y - 3 = 0 \]
Combining like terms:
\[ -3x - 4y - 5 = 0 \]
Multiplying through by -1 gives us:
\[ 3x + 4y + 5 = 0 \]

### Step 3: Use the distance formula from a point to a line
The distance \( d \) from a point \( (x_0, y_0) \) to the line \( ax + by + c = 0 \) is given by:
\[ d = \frac{|ax_0 + by_0 + c|}{\sqrt{a^2 + b^2}} \]
In our case, the line is \( 3x + 4y + 5 = 0 \), so \( a = 3 \), \( b = 4 \), and \( c = 5 \). The point is \( (1, -2) \).

### Step 4: Substitute the values into the distance formula
Substituting \( (x_0, y_0) = (1, -2) \):
\[ d = \frac{|3(1) + 4(-2) + 5|}{\sqrt{3^2 + 4^2}} \]
Calculating the numerator:
\[ 3(1) + 4(-2) + 5 = 3 - 8 + 5 = 0 \]
So, the absolute value is:
\[ |0| = 0 \]
Calculating the denominator:
\[ \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
Thus, the distance is:
\[ d = \frac{0}{5} = 0 \]

### Conclusion
The distance of the point (1, -2) from the common chord is \( 0 \). This means that the point lies on the common chord.

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