Parametric Loci Conversion and Intersection Analysis

00:00 8 marks

Consider the complex number locus defined by \(|z – (3+2i)| = |z + (1-i)|\). This equation represents the set of points in the complex plane equidistant from the points \(3+2i\) and \(-1+i\).

(a) Convert this locus into its equivalent parametric form by expressing the real part \(x\) and the imaginary part \(y\) as functions of a parameter \(t\).

(b) Given a second curve defined by the circle \((x-2)^2+(y-1)^2=4\), analyze the intersection points between the parametric line obtained in part (a) and this circle. Provide the coordinates of the intersection points, if any.

Part a

5 minutes 4 marks

Question

Given the equation \(|z – (3+2i)| = |z + (1-i)|\), let \(z=x+iy\). By equating the squares of the moduli, show that the locus reduces to a linear equation. Then, express the line in parametric form by letting \(x=t\) and deriving \(y\) in terms of \(t\).

Hint Hint

Hint

Start by setting \(|z – (3+2i)|^2 = |z + (1-i)|^2\) and expand using Cartesian form \(z=x+iy\). Simplify to find the linear equation and then parameterize by setting \(x=t\).

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Complete Answer

The locus simplifies to \(8x+6y=11\), and a parametric form is \(x=t,\quad y=\frac{11-8t}{6}\) for \(t \in \mathbb{R}\).

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Part b

5 minutes 4 marks

Question

Using the parametric form obtained in part (a), \(\left\{x=t,\; y=\frac{11-8t}{6}\right\}\), substitute these expressions into the circle equation \((x-2)^2+(y-1)^2=4\) and solve for the parameter \(t\). Then, provide the coordinates of any intersection points between the line and the circle.

Hint Hint

Hint

Substitute \(x=t\) and \(y=\frac{11-8t}{6}\) into \((x-2)^2+(y-1)^2=4\). Simplify the equation to a quadratic in \(t\) and solve for \(t\).

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Complete Answer

The intersection parameters are \(t = \frac{56 \pm \sqrt{2511}}{50}\). The intersection points are \(\left(\frac{56 \pm \sqrt{2511}}{50},\; \frac{11-8\left(\frac{56 \pm \sqrt{2511}}{50}\right)}{6}\right)\).

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