The distance of the point (1, 2) from the common chord of the circles `x^2+y^2-2x+3y-5=0` and `x^2+y^2+10x+8y=1` is

Options

A. 4
B. 3
C. 2
D. 1

Answer

To find the distance of the point (1, 2) from the common chord of the circles given by the equations \(x^2 + y^2 - 2x + 3y - 5 = 0\) and \(x^2 + y^2 + 10x + 8y = 1\), we will follow these steps:

### Step 1: Write the equations of the circles
The first circle is given by:
\[ S_1: x^2 + y^2 - 2x + 3y - 5 = 0 \]
The second circle is given by:
\[ S_2: x^2 + y^2 + 10x + 8y - 1 = 0 \]

### Step 2: Find the equation of the common chord
The equation of the common chord can be found using the formula \(S_1 - S_2 = 0\).

Substituting \(S_1\) and \(S_2\):
\[ (x^2 + y^2 - 2x + 3y - 5) - (x^2 + y^2 + 10x + 8y - 1) = 0 \]

This simplifies to:
\[ -2x + 3y - 5 - 10x - 8y + 1 = 0 \]
Combining like terms:
\[ -12x - 5y - 4 = 0 \]
Multiplying through by -1 gives:
\[ 12x + 5y + 4 = 0 \]

### Step 3: Use the distance formula
The distance \(d\) from a point \((x_0, y_0)\) to a line \(Ax + By + C = 0\) is given by:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
For our line \(12x + 5y + 4 = 0\), we have:
- \(A = 12\)
- \(B = 5\)
- \(C = 4\)

We want to find the distance from the point \((1, 2)\):
- \(x_0 = 1\)
- \(y_0 = 2\)

### Step 4: Substitute into the distance formula
Substituting into the formula:
\[ d = \frac{|12(1) + 5(2) + 4|}{\sqrt{12^2 + 5^2}} \]

Calculating the numerator:
\[ 12(1) + 5(2) + 4 = 12 + 10 + 4 = 26 \]
So, the absolute value is:
\[ |26| = 26 \]

Calculating the denominator:
\[ \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \]

### Step 5: Final calculation of distance
Now substituting back into the distance formula:
\[ d = \frac{26}{13} = 2 \]

Thus, the distance of the point (1, 2) from the common chord of the circles is:
\[ \boxed{2} \]