Tangents, normals and lociEdexcel International A Level Further Maths: Flashcards

What these 13 flashcards ask

Gradient of the ellipse \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 at (x1,y1)?
Gradient of the hyperbola \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 at (x1,y1)?
Tangent to the ellipse at (x1,y1)?
Tangent to the hyperbola at (x1,y1)?
Relation between tangent and normal gradients?
Parametric form of an ellipse?
Parametric form of a hyperbola?
Tangent to the ellipse at (a\cos\theta,b\sin\theta)?
Condition for y=mx+c to touch \frac{x^2}{a^2}+\frac{y^2}{b^2}=1?
Condition for y=mx+c to touch \frac{x^2}{a^2}-\frac{y^2}{b^2}=1?
How do you derive a tangent condition?
What is a locus?
First step in a distance locus problem?

Exam questions on Tangents, normals and loci

The ellipse $E$ has equation $\frac{x^2}{25}+\frac{y^2}{9}=1$ and $P$ is the point $\left(4,\frac95\right)$ on $E$ .
The normal to $E$ at $P$ meets the $x$ -axis at $A$ . Find the $x$ -coordinate of $A$ .2 marks
The hyperbola $H$ has equation $\frac{x^2}{16}-\frac{y^2}{9}=1$ .
Show that no tangent to $H$ has gradient $\frac12$ .2 marks
The ellipse $E$ has equation $\frac{x^2}{20}+\frac{y^2}{5}=1$ and $P$ is the point $(2,2)$ on $E$ .
Find an equation of the normal to $E$ at $P$ .3 marks

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).