Complex Numbers and Quadratic Equations · Question 1

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

Answer

(2) 2

Solution

[figure]

|z-(3+5i)|+|z-(5+11i)|=4 5: sum of distances from S(3, 5) and S'(5, 11) is constant

ellipse with foci S, S' and 2a=4 5 a2=20

Centre C = mid point of SS' =(4, 8)

2c=SS'= 22+62=2 10 c2=10

b2=a2-c2=20-10=10 b= 10

|z-(4+8i)|= 10: circle with centre C and radius 10=b

every point of an ellipse is at a distance b from its centre, with equality only at the ends of the minor axis

the circle touches the ellipse only at the two ends of the minor axis, (1, 9) and (7, 7)

number of values of z =2

Answer: (2)

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