Question

Write an equation of the parabola with the given characteristics. focus: \(\left(\frac{2}{3}, 0\right)\) directrix: \(x=-\frac{2}{3}\)

Short answer

The equation of the parabola is \(x=\frac{3}{8}y^2 + \frac{2}{3}\).

Step-by-step solution

Determine the Vertex

By examination, the focus \(\left(\frac{2}{3}, 0\right)\) and the directrix \(x=-\frac{2}{3}\) are horizontally oriented. Since the vertex is equidistant from the focus and directrix, the x-coordinate of the vertex will be the average of the x-coordinate of the focus and the x-coordinate of the directrix. So, \(h=\frac{\frac{2}{3}-\left(-\frac{2}{3}\right)}{2}= \frac{2}{3}\).

Determine the Coefficient a

Knowing that the focus is \(\frac{2}{3}\) units away from the vertex, we can calculate the focal length f as \(\frac{2}{3}\) units. We can then calculate a, as a is equal to the inverse of four times the focal length, or \(a = \frac{1}{4f} = \frac{1}{4*\frac{2}{3}} = \frac{3}{8}\).

Write down the parabola's equation

Now, we can write down the equation of the parabola, which, in this case, will be of the form \(x=a(y-k)^2 + h\). As the given focus and directrix provide no y coordinate, and the parabola opens to the right, k is zero. Thereby, substituting the values from Steps 1 and 2, the equation is \(x=\frac{3}{8}y^2 + \frac{2}{3}\).

Key concepts

Focus of a Parabola

The focus of a parabola is a specific point from which the parabola is 'equidistant' to all points on the curve and its directrix. This unique property means that if you take any point on the parabola, the distance to the focus is the same as the distance to the directrix, which is a straight line. Think of the focus as one of the central parts of a parabola's definition.

In our exercise, the focus is given as \(\left(\frac{2}{3}, 0\right)\). This position helps determine the parabola's orientation and shape. In fact, the axis of symmetry of the parabola will pass through the focus and perpendicular to the directrix.

Directrix of a Parabola

Every parabola is partnered with a directrix, a fixed line that, together with the focus, defines the shape of the parabola. For points on the parabola, the distance to the focus is equal to the distance to the directrix. This relationship maintains the symmetric nature of the parabola.

The directrix provided in the problem is \(x=-\frac{2}{3}\). Since the directrix is vertical, the parabola opens horizontally. The -2/3 value also helps in calculating the values needed to reach our final equation, proving pivotal in parabola problems.

Vertex of a Parabola

The vertex is arguably the most significant point on the parabola; it's where the parabola turns and is also the midpoint between the focus and directrix. This holds true for both vertically and horizontally oriented parabolas. It's important to note that the vertex is located at the point \(h, k\) in the parabola's standard equation \(y=a(x-h)^2+k\) or \(x=a(y-k)^2+h\).

In the worked example, using the focus and directrix, the vertex is determined to be \(\frac{2}{3}, 0\), indicating this is a horizontally oriented parabola. Notably, the 'horizontal' aspect influences the equation form chosen to describe the parabola.

Focal Length of a Parabola

Focal length, often denoted as \(f\), is the distance between the vertex and the focus of the parabola. It's directly related to the curvature of the parabola: a larger focal length results in a 'wider' and 'shallower' parabola, whereas a smaller focal length creates a 'narrower' and 'deeper' curve.

In the problem at hand, the focal length is \(f=\frac{2}{3}\) units. Knowing the focal length allows us to find coefficient \(a\), which dictates the parabola's openness. Specifically, the relationship \(a=\frac{1}{4f}\) is used for horizontal parabolas. As such, with \(f\) determined, we compute \(a=\frac{3}{8}\), directly leading us to the correct form of the parabola's equation.