Problem 29 - Algebra and Trigonometry

Question

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Directrix \(x=2\)

Step-by-Step Solution

Answer

The equation is \( x = \frac{-1}{8}y^2 \).

Step 1: Review Parabola and Directrix Basics

A parabola is a curve where each point is equidistant from a focus and a directrix. The standard equation for a parabola is either vertical, \[ y = ax^2 + bx + c \]or horizontal \[ x = ay^2 + by + c \]. If the vertex is at the origin and the directrix is vertical, we use the horizontal form.

Step 2: Identify the Form of the Equation

The directrix is a vertical line, given by \( x = 2 \), indicating a horizontal parabola. Therefore, the equation is of the form:\[ x = ay^2 \].

Step 3: Determine the Orientation and Value of ‘a’

Since the directrix is to the right of the vertex (origin), the parabola opens to the left. The distance from the vertex to the directrix is \( p = 2 \), so the parameter \( a \) is given by \( a = \frac{-1}{4}\times p \).

Step 4: Calculate the Value of 'a'

Substituting \( p = 2 \) into the formula gives:\[ a = \frac{-1}{4 \times 2} = \frac{-1}{8} \].

Step 5: Write the Final Equation

Substitute the found \( a \) into the horizontal form of the parabola equation:\[ x = \frac{-1}{8}y^2 \].

Key Concepts

Vertex at the Origin

The vertex of a parabola is a crucial point that serves as a sort of "corner" or "turning point" on the curve. When the vertex is at the origin, it is positioned at the coordinate point (0,0) in the Cartesian plane. This simplifies the equation of the parabola because there is no need to adjust for lateral or vertical shifts.
In mathematical terms, having a vertex at the origin means we can skip over any linear or constant terms that would normally move the parabola along the y or x axes. The vertex provides a fixed reference point for calculating other attributes such as the focus and directrix.
With the vertex at (0,0), the basic forms of parabola equations become:

• Vertical parabolas: \( y = ax^2 \)
• Horizontal parabolas: \( x = ay^2 \)

Each of these equations represents a perfect symmetry around the origin.

Directrix

The directrix of a parabola is a line that works alongside the focus to define its shape. In essence, a parabola consists of points that maintain an equal distance from both the directrix and the focus.
For a horizontal parabola, the directrix is a vertical line. In the provided problem, the directrix is specified as \( x = 2 \). The directrix helps in establishing the direction and orientation of the parabola.
If the directrix is on the right side of the vertex, the parabola opens to the left, and vice versa. This is because the parabola always opens away from the directrix, creating a curve toward the opposite side.
Knowing the equation of the directrix is essential when forming the parabola's equation, as it indirectly helps calculate the parameter \( a \), which defines the compression or expansion of the curve.

Horizontal Parabola

A horizontal parabola means that the parabola opens sideways rather than up or down. In mathematical terms, its equation form is \( x = ay^2 \). This form is different from the more widely recognized vertical form \( y = ax^2 \), which opens upwards or downwards.
For a horizontal parabola, the orientation relies on the positioning of the directrix and focus relative to the vertex. In our case, with a directrix at \( x = 2 \), the parabola opens to the left because the directrix is positioned to the right of the vertex at the origin. This is determined by the need for the parabola to maintain its equidistance property from the directrix and a hypothetical focus towards the left.
Horizontally oriented parabolas are less commonly encountered in basic mathematics but are foundational in advanced geometry and applications where directional curves are required.

Distance from Vertex to Directrix

The distance from the vertex to the directrix is a significant factor in determining the specific measurements of a parabola's equation. This distance is denoted by \( p \), and it directly influences the 'openness' or 'tightness' of a parabola's curve.
For the exercise at hand, the directrix \( x = 2 \) is 2 units away from the vertex at the origin. Hence, \( p = 2 \). This value is pivotal in calculating the parabola's parameter \( a \).
The parameter \( a \) can be computed using the formula \( a = \frac{-1}{4p} \), which is derived from the geometric relationship involving the parabola, focus, and directrix. Substituting \( p = 2 \) yields \( a = \frac{-1}{8} \). This results in a less "steep" curve as compared to a parabola with a smaller \( p \) value.
Understanding this relationship assists learners in transitioning from the geometric properties of parabolas to their algebraic expressions effectively.