Final answer:
The equation of the circle is (x - 1)^2 + (y + 1)^2 = 25, with its center at the intersection of the given lines (1, -1) and a radius of 5 determined by the distance from the center to the point (-2, -5).
Step-by-step explanation:
To find the equation of the circle described by the equations y = 3x - 4 and 2x + y = 1 that has diameters along these lines, we first need to find their intersection which will be the center of the circle. Solving the system of equations for x and y gives us the center of the circle.
- Replace y in the second equation with the expression from the first equation, 3x - 4:
2x + (3x - 4) = 1. - Simplify and solve for x:
5x - 4 = 1
5x = 5
x = 1. - Substitute x = 1 into the first equation to find y:
y = 3(1) - 4
y = -1. - The center of the circle is (1, -1).
- Now calculate the radius using the distance formula with the center (1, -1) and the given point (-2, -5) on the circle:
r = √[(-2 - 1)² + (-5 - (-1))²]
r = √(9 + 16)
r = √25
r = 5. - Finally, write the equation of the circle using the center-radius form:
(x - 1)² + (y + 1)² = 5².
The equation of the circle is (x - 1)² + (y + 1)² = 25.