A line meeting a complex locus

Choose your method, connect the parts and explain your conclusions. These original questions include written reasoning and sketches as well as exact answer checks.

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Before you start

Complex loci. Read the relevant method, then decide which parts you can solve without hints.

Check a misconception first · Try a connected problem

Read an original example question

The locus |z| = 3 meets the line Re(z) = 3.

Part a

Derive an equation for the imaginary part y at an intersection.

Part b

State the number of intersections.

Part c

For Re(z) = c, classify all possible numbers of intersections.

Model solution and review criteria

Part a

Writing z = 3 + iy gives $y^{2}$ = 9 − 9.

Translate modulus into $x^{2}$ + $y^{2}$ = $r^{2}$ .
Substitute the line condition.

Part b

$y^{2}$ = 0 gives the single point z = 3.

Use the sign of $y^{2}$ .
Recognise a tangent when it is zero.

Part c

There are two when |c| < 3, one when |c| = 3, and none when |c| > 3.

Include negative c through its modulus.
State the tangent and no-intersection cases.

Compare your own reasoning. No automatic examination marks are awarded.