Edexcel IAL Mathematics FP3.2.3 conic tangents and normals

Practise deriving tangents and normals to ellipses and hyperbolas, then using them to find contacts, intersections and areas.

Exam points

use implicit or parametric differentiation to prove a tangent equation at P
apply the tangent condition for y = mx + c to find tangents through a point
find the point of contact by substituting a tangent back into the conic

[Maximum number: 3]

The ellipse E has equation

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

Using calculus, show that an equation for $l_{1}$ is

2 x \cos \theta+3 y \sin \theta=6

The line $l_{2}$ passes through the origin and is perpendicular to $l_{1}$
The line $l_{1}$ intersects the line $l_{2}$ at the point Q

For $(x,y)=(3\cos\theta,2\sin\theta)$ ,
$\frac{dx}{d\theta}=-3\sin\theta,\qquad \frac{dy}{d\theta}=2\cos\theta.$
Thus
$\frac{dy}{dx}=\frac{2\cos\theta}{-3\sin\theta}.$
The tangent is
$y-2\sin\theta=\frac{2\cos\theta}{-3\sin\theta}(x-3\cos\theta),$
which rearranges to
$2x\cos\theta+3y\sin\theta=6.$

M1 M1 A1

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FP3.2.3 - Tangents and normals to these question 1