Edexcel A-Level Mathematics A2 Fp3 2 3 Tangents and Normals to These Questions

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

The point P lies on E and has coordinates $(3\cos\theta,2\sin\theta)$ .

The line $l_1$ is the tangent to E at the point P

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

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

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

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Edexcel A-Level Mathematics A2 Fp3 2 3 Tangents and Normals to These Questions question 1