Parabola
The first conic — a curve every point of which is balanced exactly between a focus and a directrix, and the source of the reflector's perfect mirror
- The focus–directrix definition of a parabola and its eccentricity \(e=1\).
- The standard equation \(y^2=4ax\) with its focus, directrix, vertex and axis, and the four standard forms.
- The latus rectum (length \(4a\)) and the focal distance \(x_1+a\).
- The parametric point \((at^2,2at)\) and the position test from \(S_1\).
- Tangents at a point, by parameter and by slope, with the condition \(c=\tfrac{a}{m}\); and normals and the chord of contact.
- Focal chords (\(t_1t_2=-1\)) and the parabola's famous reflection property.
Definition of a Parabola
Slice a cone with a plane parallel to its slant side and the cut is a parabola — one of the conic sections. Without the cone, the same curve is defined in the plane as a locus: the set of points equidistant from a fixed point, the focus, and a fixed line, the directrix. The ratio of those two distances, the eccentricity, is exactly \(e=1\).
The Standard Equation
Put the vertex at the origin, the focus at \((a,0)\) and the directrix at \(x=-a\). Equating the distance to the focus with the distance to the directrix, \(\sqrt{(x-a)^2+y^2}=x+a\), and squaring, the cross terms cancel and the standard equation drops out.
Vertex \((0,0)\), focus \((a,0)\), directrix \(x=-a\), axis the \(x\)-axis. The curve opens toward the focus and is symmetric about its axis. A shifted parabola \((y-k)^2=4a(x-h)\) is the same shape with vertex moved to \((h,k)\).
The Four Standard Forms
By choosing which axis the parabola opens along, and in which direction, there are four standard equations. The sign and the variable that is squared tell you everything about the orientation.
If \(y\) is squared the axis is horizontal; if \(x\) is squared it is vertical.
A \(+\) opens right or up (toward positive); a \(-\) opens left or down.
Always at distance \(a\) from the vertex, on opposite sides along the axis.
Latus Rectum & Focal Distance
Two measurements show up everywhere. The latus rectum is the focal chord perpendicular to the axis — it fixes how "wide" the parabola is. The focal distance of a point is its distance to the focus, and on a parabola it has an especially simple form.
For \(y^2=4ax\) the latus rectum runs from \((a,2a)\) to \((a,-2a)\), length \(4a\). The focal distance equals \(x_1+a\) directly from the definition — the distance to the directrix \(x=-a\) is also \(x_1+a\).
Parametric Form & Position of a Point
A single parameter \(t\) sweeps out the whole parabola, which makes chords, tangents and focal-chord conditions remarkably clean. And substituting a point into \(S\equiv y^2-4ax\) tells you at once where it lies.
The point \((at^2,2at)\) lies on \(y^2=4ax\) for every \(t\). For the position test, \(S_1<0\) means the point is inside (on the focus side), \(S_1=0\) means it is on the curve, and \(S_1>0\) means it is outside.
Tangents
As with the circle, replacing \(y^2\to yy_1\) and \(x\to\tfrac{x+x_1}{2}\) in \(y^2=4ax\) gives the tangent at a point. There are three equivalent ways to write a tangent, picked to match the data you hold.
The first is the tangent at a point \((x_1,y_1)\); the second is the tangent at parameter \(t\); the third is the tangent of slope \(m\), touching at \(\left(\tfrac{a}{m^2},\tfrac{2a}{m}\right)\). A line \(y=mx+c\) is tangent precisely when \(c=\tfrac{a}{m}\).
Normals & the Chord of Contact
The normal is perpendicular to the tangent at the point of contact. The chord of contact from an external point joins the two points where its tangents touch — and, just like the circle, it is the equation \(T=0\).
The first is the normal at \((x_1,y_1)\) (slope \(-\tfrac{y_1}{2a}\)); in parameter form it is \(y+tx=2at+at^3\). The second, \(T=0\), serves as the tangent at a point, the chord of contact from an external point, and — set equal to \(S_1\) — the chord with midpoint \((x_1,y_1)\).
Focal Chords & the Reflection Property
A chord through the focus is a focal chord. In parameter form it satisfies a single elegant relation, and the parabola's signature optical behaviour follows from the geometry of its tangents.
The ends \((at_1^2,2at_1)\) and \((at_2^2,2at_2)\) of a focal chord satisfy \(t_1t_2=-1\); consequently the tangents at the two ends are perpendicular and meet on the directrix. The reflection property — any ray parallel to the axis reflects through the focus — is what makes parabolic dishes and headlamps work.
Putting It to Work
Problem. Find the vertex, focus, directrix and latus rectum of \(y^2=12x\).
Solution. Comparing with \(y^2=4ax\) gives \(4a=12\), so \(a=3\):
Problem. Find the vertex, focus and directrix of \(y^2-4y-8x+28=0\).
Solution. Complete the square in \(y\):
Problem. Find the focal distance of the point \((2,4)\) on \(y^2=8x\).
Solution. Here \(4a=8\), so \(a=2\); the focal distance is \(x_1+a\):
Problem. Find the tangent to \(y^2=8x\) at the point \((2,4)\).
Solution. With \(2a=4\), the tangent \(yy_1=2a(x+x_1)\) gives:
Problem. Find the tangent to \(y^2=8x\) with slope \(2\), and its point of contact.
Solution. Use \(y=mx+\tfrac{a}{m}\) with \(a=2,\ m=2\), contact \(\left(\tfrac{a}{m^2},\tfrac{2a}{m}\right)\):
Problem. Find the normal to \(y^2=12x\) at the point \((3,6)\).
Solution. With \(2a=6\), the normal has slope \(-\tfrac{y_1}{2a}=-1\):
Chapter Summary
Equidistant from focus and directrix; eccentricity \(e=1\).
\(y^2=4ax\): vertex \((0,0)\), focus \((a,0)\), directrix \(x=-a\).
Latus rectum \(4a\); focal distance \(x_1+a\).
Point \((at^2,2at)\); position from \(S_1=y_1^2-4ax_1\).
Tangent \(y=mx+\tfrac{a}{m}\); \(T=0\) for tangent / chord of contact; chord-with-midpoint \(T=S_1\).
\(t_1t_2=-1\); rays parallel to the axis reflect through the focus.
Problems
Compare with the matching standard form first to read off \(a\), then apply the focal, tangent or normal result the question calls for. Difficulty rises down the list.
- Find the focus and directrix of \(y^2=16x\).
- Find the equation of the parabola with vertex at the origin and focus \((0,2)\).
- Find the length of the latus rectum of \(x^2=-10y\).
- Find the vertex, focus and directrix of \((y+1)^2=8(x-2)\).
- Find the focal distance of the point \((4,8)\) on \(y^2=16x\).
- Find the coordinates of the point on \(y^2=12x\) whose parameter is \(t=-2\).
- Find the tangent to \(y^2=4x\) at the point \((1,2)\).
- Find the tangent to \(y^2=16x\) having slope \(2\).
- Determine whether the point \((2,3)\) lies inside or outside \(y^2=8x\).
- Find the normal to \(y^2=8x\) at the point \((2,4)\).
- The ends of a focal chord of \(y^2=4ax\) have parameters \(t_1\) and \(t_2\). Show that \(t_1t_2=-1\).
- Find the equation of the chord of \(y^2=8x\) which is bisected at the point \((4,2)\).