State the focus-directrix property of a parabola.

Every point on the parabola is the same distance from the focus as from the directrix.

What is the distance from P(x,y) on y^2=4ax to the focus?

x+a, the distance to the directrix.

What are the asymptotes of xy=c^2?

The coordinate axes, x=0 and y=0.

What condition makes y=mx+c a tangent to y^2=4ax?

c=\frac am, from setting the discriminant of the quadratic in x to zero.

How do you find the equation of a locus?

Let P=(x,y), write the condition using distances, square and simplify.

Parabola and rectangular hyperbolaEdexcel A-Level Further Maths: Flashcards

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State the Cartesian and parametric equations of the parabola with focus $(a,0)$.

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State the Cartesian and parametric equations of the parabola with focus $(a,0)$ .: $y^2=4ax$ ; $x=at^2$ , $y=2at$ .
State the focus and directrix of $y^2=4ax$ .: Focus $(a,0)$ ; directrix $x=-a$ .
Find the focus of $y^2=20x$ .: $4a=20$ so $a=5$ : focus $(5,0)$ .
State the focus-directrix property of a parabola.: Every point on the parabola is the same distance from the focus as from the directrix.
What is the distance from $P(x,y)$ on $y^2=4ax$ to the focus?: $x+a$ , the distance to the directrix.
State the Cartesian and parametric equations of the rectangular hyperbola.: $xy=c^2$ ; $x=ct$ , $y=\frac ct$ .
What are the asymptotes of $xy=c^2$ ?: The coordinate axes, $x=0$ and $y=0$ .
State $\frac{dy}{dx}$ for $y^2=4ax$ in terms of $y$ and in terms of $t$ .: $\frac{2a}{y}=\frac1t$ .
State $\frac{dy}{dx}$ for $xy=c^2$ at $\left(ct,\frac ct\right)$ .: $-\frac1{t^2}$ .
State the tangent to $y^2=4ax$ at $(at^2,2at)$ .: $ty=x+at^2$ .
State the normal to $y^2=4ax$ at $(at^2,2at)$ .: $y+tx=2at+at^3$ .
State the tangent to $xy=c^2$ at $\left(ct,\frac ct\right)$ .: $x+t^2y=2ct$ .
What condition makes $y=mx+c$ a tangent to $y^2=4ax$ ?: $c=\frac am$ , from setting the discriminant of the quadratic in $x$ to zero.
How do you find the equation of a locus?: Let $P=(x,y)$ , write the condition using distances, square and simplify.

Exam questions on Parabola and rectangular hyperbola

A parabola $C$ has equation $y^2=20x$ .
The point $P$ on $C$ has $x$ -coordinate $20$ . Use the focus-directrix property to find the distance from $P$ to the focus.2 marks
The rectangular hyperbola $H$ has equation $xy=36$ .
Find the equation of the normal to $H$ at the point $(4,9)$ , giving your answer in the form $ay=bx+c$ where $a$ , $b$ and $c$ are integers.2 marks
The parabola $C$ has equation $y^2=8x$ .
The line $y=mx+2$ is a tangent to $C$ . Find the value of $m$ .3 marks

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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).