Pearson Edexcel IAL Mathematics FP3.2.4 Simple loci problems

Practise forming simple loci from moving points on conics by expressing coordinates in a parameter and eliminating it.

Exam points

eliminate θ or k from point coordinates to show a stated locus equation
use sin²θ + cos²θ = 1 to convert parametric coordinates into an ellipse

[Maximum number: 3]

The ellipse E has equation

\frac{x^{2}}{9}+\frac{y^{2}}{4}=1

Show that, as $\theta$ varies, the point Q lies on the curve with equation

\left(x^{2}+y^{2}\right)^{2}=\alpha x^{2}+\beta y^{2}

where $\alpha$ and $\beta$ are constants to be determined.

For the point Q,
$\frac{y}{x}=\frac32\tan\theta.$
Using the result from the preceding parts to eliminate $\theta$ gives
$(x^2+y^2)^2=9x^2+4y^2.$
Therefore
$\alpha=9,\qquad \beta=4.$

M1 M1 A1

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FP3.2.4 - Simple loci problems question 1