Pearson Edexcel IAL Mathematics FP1.4.1 Cartesian equations for the parabola

Practise reading Cartesian parabola forms and using focus, directrix and distance information to state constants or key coordinates.

Exam points

read a or focus coordinates from y^2=4ax, then use them to locate the directrix
use focus-directrix or tangent information to form the Cartesian equation of a parabola

[Maximum number: 4]

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 coordinates of Q

Given that
- the point R is the focus of C
- the line l is the directrix of C
- the line through Q and R meets l at the point S

At (8,2), $c^2=16$ , so c=4, and ct=8, so t=2.
Normal: 8x-2y=4(16-1)=60, so y=4x-30.
With $y^2=6x$ : $(4x-30)^2=6x$ .
$16x^2-246x+900=0$ $x=6\text{ or }\frac{75}{8}$
Since y>0, $Q=\left(\frac{75}{8},\frac{15}{2}\right)$ .
B1 M1 dM1 A1

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FP1.4.1 - Cartesian equations for the parabola question 1