The number of values of z C , satisfying the equations |z-(4+8i)|= 10 and |z-(3+5i)|+|z-(5+11i)|=4 5 , is:

Step-by-step solution

The first equation |z-(4+8i)|= 10 represents a circle with center C(4, 8) and radius r = 10 . The second equation |z-(3+5i)|+|z-(5+11i)|=4 5 represents an ellipse with foci at S₁(3, 5) and S₂(5, 11) , and length of the major axis 2a = 4 5 . The distance between the foci is 2ae = (5-3)^2 + (11-5)^2 = 4+36 = 40 = 2 10 . The center of the ellipse is the midpoint of the line segment joining the foci, which is ( 3+5 2 , 5+11 2 ) = (4, 8) . This is the same as the center of the circle. The semi-major axis is a = 2 5 , so