The circle equation |z - (a + ib)| = r has its centre at which point?

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Boost your skills in A Level Further Mathematics Core Pure. Study with flashcards and multiple choice questions, each question is followed by hints and explanations. Get prepared for your exam with confidence!

Multiple Choice

The circle equation |z - (a + ib)| = r has its centre at which point?

Explanation:
The distance interpretation is key here: in the complex plane, the modulus |z − z0| is the distance from z to the fixed point z0. So |z − (a + ib)| = r is the set of all points z whose distance to the fixed point a + ib is r. That means the circle is centered at a + ib with radius r. If you write z as x + iy, you get sqrt[(x − a)^2 + (y − b)^2] = r, which shows the center is at the point (a, b) in the plane, i.e., the complex number a + ib.

The distance interpretation is key here: in the complex plane, the modulus |z − z0| is the distance from z to the fixed point z0. So |z − (a + ib)| = r is the set of all points z whose distance to the fixed point a + ib is r. That means the circle is centered at a + ib with radius r. If you write z as x + iy, you get sqrt[(x − a)^2 + (y − b)^2] = r, which shows the center is at the point (a, b) in the plane, i.e., the complex number a + ib.

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