Parabolas and rectangular hyperbolas

What you’ll learn

• Use Cartesian and parametric forms of basic conics.
• Find tangents, normals and intersections.

Understand the idea

The parabola $y^2=4ax$ has focus $(a,0)$ and directrix $x=-a$. The parameterisation $(at^2,2at)$ places every real value of $t$ on the curve and often makes tangent calculations shorter.

A rectangular hyperbola $xy=c^2$ has the coordinate axes as asymptotes and can be parameterised by $(ct,c/t)$, where $t\ne0$. Differentiate implicitly or parametrically to obtain a gradient, then use the point-gradient equation; solve simultaneous equations to investigate a line's intersections with a conic.

Key formulae

$$y^2=4ax,\qquad(x,y)=(at^2,2at)$$

$$ty=x+at^2$$

$$xy=c^2,\qquad(x,y)=(ct,c/t)$$

$$\frac{dy}{dx}=-\frac{y}{x}\quad(xy=c^2)$$

Worked example

Find the tangent to $y^2=8x$ at parameter $t=3$.

1. Here $4a=8$, so $a=2$.
2. The point is $(at^2,2at)=(18,12)$.
3. The tangent formula gives $3y=x+2(3^2)$.

Answer: $x-3y+18=0$.