Question
Question: If |z – 5 – 7i| = 9, then find the greatest and least values of |z – 2 – 3i|....
If |z – 5 – 7i| = 9, then find the greatest and least values of |z – 2 – 3i|.
Answer
Greatest value = 14, Least value = 4
Explanation
Solution
The equation |z - 5 - 7i| = 9 represents a circle centered at C₁(5, 7) with radius R=9. We need to find the range of distances from points z on this circle to a fixed point P₂(2, 3), represented by |z - 2 - 3i|. The distance between the center C₁ and the fixed point P₂ is d = |(5+7i) - (2+3i)| = |3+4i| = √(3²+4²) = 5.
The greatest distance is R+d = 9+5 = 14.
The least distance is R-d = 9-5 = 4.