Edexcel A-Level Mathematics AS Fp1 4 1 Cartesian Equations for the Parabola Questions

Practise Edexcel IAL FP1.4.1 by using Cartesian conic equations, focus-directrix geometry, key coordinates and asymptotes.

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
state asymptotes of a rectangular hyperbola from its Cartesian equation

[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, $Q=\left(\frac{75}{8},\frac{15}{2}\right)$ .

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Edexcel A-Level Mathematics AS Fp1 4 1 Cartesian Equations for the Parabola Questions question 1