Pearson Edexcel IAL Mathematics FP1.4.2 Parametric equations for conics

Practise using parametric points on parabolas and rectangular hyperbolas to represent coordinates before forming equations or constraints.

[Maximum number: 5]

The rectangular hyperbola H has equation $x y=c^{2}$ where c is a positive constant.

The point $P\left(c t, \frac{c}{t}\right)$ , where t>0, lies on H

determine the exact length of QS

For $y^2=6x$ , 4a=6, so focus $R=\left(\frac32,0\right)$ and directrix $x=-\frac32$ .
Line through $Q\left(\frac{75}{8},\frac{15}{2}\right)$ and $R\left(\frac32,0\right)$ has gradient $\frac{20}{21}$ .
$y=\frac{20}{21}\left(x-\frac32\right)$
At $x=-\frac32$ , $y=-\frac{20}{7}$ , so $S=\left(-\frac32,-\frac{20}{7}\right)$ .
$QS=\sqrt{\left(\frac{75}{8}+\frac32\right)^2+\left(\frac{15}{2}+\frac{20}{7}\right)^2}=\frac{2925}{56}$
B1 M1 M1 M1 A1

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FP1.4.2 - Parametric equations for conics question 1