Pearson Edexcel IAL Mathematics FP1.4.4 Tangents & normals to conics

Practise finding tangents and normals to parabolas and rectangular hyperbolas, then using those lines to solve intersections and constants.

Exam points

differentiate a conic equation to form a tangent or normal at a given point
use a normal equation with another conic to find the second intersection point
apply line intercepts or triangle area conditions to determine a constant

[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

Use calculus to show that an equation of the normal to H at P is

t^{3} x-t y=c\left(t^{4}-1\right)

The parabola C has equation $y^{2}=6 x$
The normal to H at the point with coordinates (8,2) meets C at the point Q where y>0

$xy=c^2\Rightarrow y=c^2x^{-1}$ $\frac{dy}{dx}=-c^2x^{-2}$
At P(ct,c/t), $\frac{dy}{dx}=-\frac{1}{t^2}$ , so the normal gradient is $t^2$ .
$y-\frac ct=t^2(x-ct)$ $ty-c=t^3x-ct^4$ $t^3x-ty=c(t^4-1)$
B1 M1 M1 A1

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FP1.4.4 - Tangents and normals to conics question 1