Problem 24 - Precalculus

Question

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Directrix: \(x=3\)

Step-by-Step Solution

Answer

The standard form of the equation of the parabola is \(x = -12y^2\).

Step 1: Determine the direction of the parabola

The given directrix is a vertical line \(x=3\). The parabola will always open away from the directrix. Thus, it will be facing left or opening leftwards and its equation will be of the form \(x=ay^2\).

Step 2: Determine the focal length

The focal length 'f' for a parabola is the distance from the vertex to the directrix. Since the vertex is at the origin and the directrix is at \(x=3\), the focal length is therefore 3 units to the left or -3.

Step 3: Determine the standard form

Using the fact that \(4af=x\), where 'a' representing the position of the directrix and 'f' the focal length of the parabola, substituting the values, we get \(-4a \cdot 3=x\), which simplifies to \(x=-12y^2\). Thus the standard form of the equation of the parabola is \(x=-12y^2\).

Key Concepts

Directrix of a Parabola

Understanding the directrix of a parabola is crucial when you're studying conic sections. The directrix is a line that, together with the focus, serves to define a parabola. Essentially, a parabola is the set of all points that are equidistant from the focus, a fixed point, and the directrix, a fixed straight line.

For the exercise at hand, the directrix is given as the vertical line with the equation \(x=3\). Because a parabola's nature is to open away from its directrix, knowing the placement of the directrix immediately tells us the parabola's orientation — in this case, it opens to the left. This is a vital piece of information as it dictates the form of the equation you're trying to find. For a vertical directrix like this one, the parabola has a horizontal axis of symmetry and its standard form equation will involve \(y^2\).

Parabola Opening Direction

The opening direction of a parabola is a clear indicator of its shape and equation. When the parabola opens to the left or right, it means we're dealing with a horizontally oriented parabola, which has its standard form equation as \(x=ay^2\) or \(x=-ay^2\) depending on whether it opens to the right or left, respectively. Conversely, when a parabola opens upwards or downwards, the standard form will include \(y=ax^2\) or \(y=-ax^2\).

In our exercise, after noting that the directrix is vertical (\(x=3\)), we inferred that the parabola must open to the left—away from the directrix. Hence, its equation takes the form \(x=-ay^2\), signaling a leftward opening with a negative coefficient indicating the direction of the opening.

Focal Length of a Parabola

The focal length of a parabola, often denoted as 'f', is the distance from the vertex to the focus. It is also half the distance between the vertex and the directrix. This concept is crucial for graphing and formulating the equation of a parabola since the value of 'f' defines the 'a' coefficient in the parabola's equation.

In the worked-out problem, the vertex of the parabola is at the origin (0,0), and the directrix is the line \(x=3\). Accordingly, the focal length is the distance from the origin to the directrix, which is 3 units. Since the parabola opens to the left (a horizontal opening), the focal length is negative, indicating that the focus is to the left of the vertex. The relationship \(4af = x\) then helps us determine the specific ‘a’ value for our equation, which turns out to be -12 when considering the direction of the opening and the focal length. The standard form of the equation becomes \(x=-12y^2\), confirming the characteristics of the parabola given in the exercise.