The locus of the complex number z satisfying |z - | = 3|z - | is a circle C with its center at z₀ = 4 + 0i and radius r = 3 . If and are real numbers such that < , then the value of ^2 + ^2 is:

Step-by-step solution

Let z = x + iy . The given equation is |x - + iy|^2 = 9|x - + iy|^2 . (x - )^2 + y^2 = 9[(x - )^2 + y^2] x^2 - 2 x + ^2 + y^2 = 9x^2 - 18 x + 9 ^2 + 9y^2 8x^2 + 8y^2 + (2 - 18 )x + (9 ^2 - ^2) = 0 Dividing by 8 , we get the standard equation of the circle: x^2 + y^2 + - 9 4 x + 9 ^2 - ^2 8 = 0 The center of this circle is (- - 9 8 , 0 ) = ( 9 - 8 , 0 ) . We are given that the center is (4, 0) , so: 9 - 8 = 4 9 - = 32 = 9 - 32 The radius r is given by r^2 = g^2 + f^2 - c . We are given r = 3 , so r^2 = 9 . (-4)^2 +