The distance from origin to the radical axis of the circles `x^2+y^2-3x+2y-4=0, x^2+y^2+x-y+1=0` is

Options

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

Answer

To find the distance from the origin to the radical axis of the given circles, we will follow these steps:

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

### Step 2: Find the radical axis
The radical axis can be found using the equation \( S_1 - S_2 = 0 \).

Calculating \( S_1 - S_2 \):
\[ S_1 - S_2 = (x^2 + y^2 - 3x + 2y - 4) - (x^2 + y^2 + x - y + 1) \]

This simplifies to:
\[ -3x + 2y - 4 - x + y - 1 = 0 \]
Combining like terms:
\[ -4x + 3y - 5 = 0 \]

Rearranging gives us the equation of the radical axis:
\[ 4x - 3y + 5 = 0 \]

### Step 3: Calculate the distance from the origin to the radical axis
The distance \( d \) from a point \( (x_0, y_0) \) to the line \( Ax + By + C = 0 \) is given by the formula:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]

For our case, the line is \( 4x - 3y + 5 = 0 \) (where \( A = 4 \), \( B = -3 \), and \( C = 5 \)), and the point is the origin \( (0, 0) \).

Substituting into the formula:
\[ d = \frac{|4(0) - 3(0) + 5|}{\sqrt{4^2 + (-3)^2}} = \frac{|5|}{\sqrt{16 + 9}} = \frac{5}{\sqrt{25}} = \frac{5}{5} = 1 \]

### Final Answer
The distance from the origin to the radical axis of the circles is \( 1 \).

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