The lines L1:2x-3y=5 and L2:3x-4y=7 are the diameters of a circle that has area: A=pi*r2=154. Determine the equation of the circle

Answer: To determine the equation of the circle, we first need to find the center and radius of the circle. We can do this by finding the point where the two diameters intersect. Let's find the intersection point of lines L1 and L2: Line L1: 2x - 3y = 5 Line L2: 3x - 4y = 7 We can use the method of substitution to solve for the intersection point. First, solve one of the equations for one variable and then substitute it into the other equation: From L1: 2x = 5 + 3y x = (5 + 3y)/2 Now, substitute this expression for x into L2: 3((5 + 3y)/2) - 4y = 7 Multiply both sides by 2 to get rid of the fraction: 3(5 + 3y) - 8y = 14 Now, distribute the 3 on the left side: 15 + 9y - 8y = 14 Combine like terms: 15 + y = 14 Subtract 15 from both sides: y = -1 Now that we have found y = -1, substitute this value back into the equation for x from L1: x = (5 + 3y)/2 x = (5 + 3(-1))/2 x = (5 - 3)/2 x = 2/2 x = 1 So, the intersection point of the two diameters is (1, -1), which is the center of the circle. Now, we need to find the radius (r) of the circle. We know that the area of the circle is given as A = πr^2 = 154. We can solve for r: πr^2 = 154 r^2 = 154/π r = √(154/π) Now, we can write the equation of the circle in standard form: (x - h)^2 + (y - k)^2 = r^2 Where (h, k) is the center of the circle and r is the radius. Plugging in the values we found: (x - 1)^2 + (y - (-1))^2 = (√(154/π))^2 (x - 1)^2 + (y + 1)^2 = 154/π So, the equation of the circle is: (x - 1)^2 + (y + 1)^2 = 154/π