Loci in Argand Diagrams (Edexcel A-Level Further Mathematics): Quizzes

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Loci in Argand Diagrams
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What geometric shape does the locus ∣z−z1∣=r|z - z_1| = r represent on an Argand diagram?

A circle

For the locus ∣z−(3+4i)∣=5|z - (3 + 4i)| = 5, what is the centre of the circle?

(3,4)(3, 4)

What does the locus ∣z−z1∣=∣z−z2∣|z - z_1| = |z - z_2| represent?

The perpendicular bisector

What is the geometric interpretation of ∣z−z1∣=∣z−z2∣|z - z_1| = |z - z_2|?

Points equidistant from z1z_1 and z2z_2

For ∣z−z1∣=k∣z−z2∣|z - z_1| = k|z - z_2|, what is the locus when k=1k = 1?

Perpendicular bisector

For ∣z−z1∣=k∣z−z2∣|z - z_1| = k|z - z_2|, what is the locus when k≠1k \neq 1?

A circle

What shape does arg⁡(z−z1)=θ\arg(z - z_1) = \theta represent?

A half-line starting from z1z_1

For the locus ∣z−(2−i)∣=3|z - (2 - i)| = 3, what is the centre?

(2,−1)(2, -1)

What is the general formula for the modulus ∣z∣|z| where z=x+yiz = x + yi?

x2+y2\sqrt{x^2 + y^2}

What does ∣z−z1∣=r|z - z_1| = r describe geometrically?

Points at constant distance rr from z1z_1

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