Let a circle C in the complex plane pass through the points z₁ = 3+4i , z₂ = 4+3i and z₃ = 5i . Let L₁ and L₂ be the tangents to C at z₁ and z₂ respectively, and let them intersect at a point z₀ . If d is the perpendicular distance from z₀ to the line passing through z₁ and z₂ , then the value of 98 d^2 is

Step-by-step solution

Let the points be z₁ = (3, 4) , z₂ = (4, 3) , and z₃ = (0, 5) . The moduli of these points are |z₁| = 3^2+4^2 = 5 , |z₂| = 4^2+3^2 = 5 , and |z₃| = 5 . Thus, the circle C is centered at the origin with radius 5 , and its equation is x^2 + y^2 = 25 . The equation of the tangent L₁ at z₁(3, 4) is 3x + 4y = 25 . The equation of the tangent L₂ at z₂(4, 3) is 4x + 3y = 25 . To find the intersection point z₀ , we solve the two tangent equations. Adding them gives 7x + 7y = 50 x + y = 50 7 . Subtracting them gives -x + y