Circle C₁ has equation $x^2 + y^2 + 6x + 10y + 9 = 0.$ The centre of circle C₂ is $(9, 11)$ - Scottish Highers Maths - Question 5 - 2015

To find the equation of circle C₃, we will define its characteristics based on the information provided:

1. Find the center of C₃: Since circles C₁, C₂, and C₃ are collinear, the center of C₃ can be calculated considering that it's along the line joining the centers of C₁ and C₂. The slope of the line between centers C₁($-3, -5$) and C₂($9, 11$) is given by: $slope = \frac{y_2 - y_1}{x_2 - x_1} = \frac{11 - (-5)}{9 - (-3)} = \frac{16}{12} = \frac{4}{3}$ Thus, the relative position of C₃ would follow this slope.

2. Use distances: Since C₁ touches C₃ internally and C₂ also touches C₃ internally, we can denote:

   • Radius of C₃ from C₁ (denote as $r_3$) will be equal to its radius plus the radius of C₁: $r_3 + 5 = 20 - r_3$ (as found earlier). Therefore: $2r_3 = 15$ This leads to: $r_3 = 7.5$.

3. Equation of C₃: With center $(x, y) = (x_{C₃}, y_{C₃})$, the radius is now known as 7.5. Using the point $(x_{C₃}, y_{C₃})$, we can calculate the exact coordinates based on collinearity and distance conditions. However, it must satisfy:

   $\frac{7.5 - 5}{r_2 - 7.5} = \frac{distance}{9 - x_{C₃}}$ Given that the exact coordinates can be determined graphically or by symmetry: The general equation for the circle would thus be: $(x-a)^2 + (y-b)^2 = (7.5)^2$ Finalizing with a specific center calculated.

The complete circle equation derivation requires substituting specific $(a, b)$ parameters based on geometric conclusions derived from the entirety of given information.