Franklin’s Blog

Tilted Parabola

2016 Dec 7

Find the equation of the graph of a parabola that is neither vertical or horizontal.

It is commonly known that a parabola is the locus of all points equidistant from a point, called the focus, and a line, called the directrix. The quadratic function represents a parabola with a horizontal directrix, and the square root function can represent half of a parabola with a vertical directrix. Given a horizontal or vertical directrix, it is easy to write the equation of the parabola, since the distance of a point on the parabola from the focus can be found by the distance formula and the distance of a point on the parabola from a horizontal or vertical line can be found by the absolute value of the difference between the abscissa or ordinate of the point on the parabola and the y- or x-value of the horizontal or vertical line. However, now that we have a formula for the distance between a point and any line with defined nonzero slope, we can create a parabola that is tilted at an angle. The way to do this would be to choose a directrix line \(y=mx+b\) and a focus \((x_0, y_0)\), and the equation of the parabola would be

\[\sqrt{(x-x_0)^2+(y-y_0)^2}=\frac{\mid y-mx-b\mid}{\sqrt{m^{2}+1}}\]

Let’s try an example: the parabola with focus \((1,1)\) and directrix \(y=-x\). First we must set up our equation:

\[\sqrt{(x-1)^2+(y-1)^2}=\frac{\mid y+x-0\mid}{\sqrt{(-1)^{2}+1}}\]
\[\sqrt{(x-1)^2+(y-1)^2}=\frac{\mid y+x\mid}{\sqrt{2}}\]

And now we must simplify. To begin, we square both sides.

\[(x-1)^2+(y-1)^2=\frac{(y+x)^2}{2}\]

Then we expand the binomials.

\[x^2-2x+1+y^2-2y+1=\frac{1}{2}y^2+\frac{1}{2}x^2+xy\]
\[\frac{1}{2}y^2-2y-xy+\frac{1}{2}x^2-2x+2=0\]
\[(\frac{1}{2})y^2+(-2-x)y+(\frac{1}{2}x^2-2x+2)=0\]

Now that we have this in \(ax^2+bx+c\) form as a quadratic of y, we can solve it for y in terms of x using the quadratic formula.

\[y=2+x\pm \sqrt{(2+x)^2-2(\frac{1}{2}x^2-2x+2)}\]
\[y=2+x\pm \sqrt{x^2+4x+4-x^2+4x-4}\]
\[y=2+x\pm \sqrt{8x}\]
\[y=2+x\pm 2\sqrt{2x}\]

In this case, \(y\) is not a function of \(x\) because of the plus-or-minus, but the parabola can be represented as the two functions \(2+x+2\sqrt{2x}\) and \(2+x-2\sqrt{2x}\) together. Here is a graph of the parabola, along with its focus and directrix:

Fig 1