To find the equation of the radical axis of the given circles, we follow these steps:
1. Write the equations of the circles in a standard form for comparison:
The first circle’s equation is given by:
[tex]\[
x^2 + y^2 - 3x - 4y + 5 = 0
\][/tex]
This is already in the standard form.
The second circle’s equation is given by:
[tex]\[
3(x^2 + y^2) - 7x + 8y - 11 = 0
\][/tex]
We simplify this equation to match the format of the first circle. Start by dividing every term by 3:
[tex]\[
x^2 + y^2 - \frac{7}{3}x + \frac{8}{3}y - \frac{11}{3} = 0
\][/tex]
2. Identify the coefficients for each circle:
For the first circle [tex]\( x^2 + y^2 - 3x - 4y + 5 = 0 \)[/tex]:
- [tex]\( a_1 = -3 \)[/tex]
- [tex]\( b_1 = -4 \)[/tex]
- [tex]\( c_1 = 5 \)[/tex]
For the second circle [tex]\( x^2 + y^2 - \frac{7}{3}x + \frac{8}{3}y - \frac{11}{3} = 0 \)[/tex]:
- [tex]\( a_2 = -\frac{7}{3} \)[/tex]
- [tex]\( b_2 = \frac{8}{3} \)[/tex]
- [tex]\( c_2 = -\frac{11}{3} \)[/tex]
3. Compute the coefficients for the radical axis using the formula:
The equation for the radical axis is obtained by subtracting the corresponding coefficients of the circles:
[tex]\[
(a_1 - a_2)x + (b_1 - b_2)y + (c_1 - c_2) = 0
\][/tex]
Calculate each coefficient for the radical axis equation:
- For [tex]\( x \)[/tex] term:
[tex]\[
a_r = a_1 - a_2 = -3 - \left(-\frac{7}{3}\right) = -3 + \frac{7}{3} = -3 + 2.333\ldots = -0.666\ldots
\][/tex]
- For [tex]\( y \)[/tex] term:
[tex]\[
b_r = b_1 - b_2 = -4 - \left(\frac{8}{3}\right) = -4 - 2.666\ldots = -6.666\ldots
\][/tex]
- For the constant term:
[tex]\[
c_r = c_1 - c_2 = 5 - \left(-\frac{11}{3}\right) = 5 + \frac{11}{3} = 5 + 3.666\ldots = 8.666\ldots
\][/tex]
4. Form the radical axis equation:
Substituting the computed coefficients into the radical axis equation, we get:
[tex]\[
-0.6667x - 6.6667y + 8.6667 = 0
\][/tex]
Thus, the equation of the radical axis for the given circles is approximately:
[tex]\[
-0.6667x - 6.6667y + 8.6667 = 0
\][/tex]
