Problem 20 - Algebra and Trigonometry

Question

In Exercises 17–30, find the standard form of the equation of each parabola satisfying the given conditions. Focus: \(\quad(-10,0) ;\) Directrix: \(x=10\)

Step-by-Step Solution

Answer

The standard form of the equation for the given parabola is \(y^2 = -40x\).

Step 1: Determine the vertex

Since the directrix is given as \(x = 10\), the x-coordinate of the vertex (h) is halfway between the focus and the directrix. Therefore, the x-coordinate of the vertex (\(h\)) can be calculated as the average \(h = (Focus_x + Directrix_x)/2 = (-10 + 10)/2 = 0\).

Step 2: Determine the value of k and p

The y-coordinate of the vertex, k, is the same as the y-coordinate of the focus, \(k = 0\). For a parabola that opens to the left or right, p is the distance from the vertex to the focus or from the vertex to the directrix, \(p = Focus_x - h = (-10 - 0) = -10\). Since this parabola opens to the left, p is negative. If it were to open to the right, p would be positive.

Step 3: Write the equation

As the parabola is horizontal in nature, the equation's structure should be \((y-k)^2 = 4p(x-h)\). By substituting the values of h=k=0 and p=-10 into this equation, the final standard form of the equation of the parabola can be given as \((y-0)^2 = 4*(-10)*(x-0)\). Simplifying this, the final equation is \(y^2 = -40x\).

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a critical point, often described as the "tip" or highest/lowest point, depending on the orientation. It's a central feature that helps define the shape and position of the parabola.
Understanding the vertex's role begins with knowing its coordinates, \(h, k\). In our exercise, the parabola's characteristics, specifically its focus and directrix, guide us in identifying these coordinates.

• The x-coordinate \(h\) of the vertex lies exactly between the focus's x-coordinate and the x-value of the directrix.
• We calculate it by averaging these x-values. For our parabola, this becomes \(h = (Focus_x + Directrix_x)/2\).
• The y-coordinate \(k\) aligns with that of the focus when the directrix is vertical as it is in this case.

In the given problem, following these guidelines yields the vertex \(0, 0\), which is central to formulating the parabola's equation.

Focus and Directrix

The focus and directrix are foundational in understanding parabola's geometry. These two components work together to define the set of all points making up the parabola.
The focus is a fixed point within the parabola from which distances to any point on the parabola are considered.

• In this problem, the focus is \((-10, 0)\).
• The directrix, \(x = 10\), is a line equidistant from the parabola as the focus.
• A point's path, keeping equal distance from the focus point and the directrix line, forms the parabolic curve.

These components establish the parabola's symmetry and direction. In a nutshell, if you imagine each point on a parabola balancing equidistance between the focus and the directrix, you'll grasp how essential these two features are in plotting the curve’s path and revealing its shape.

Standard Form of a Parabola

The standard form of a parabola is essentially its equation, providing a clear-cut way to represent the curve mathematically. For parabolas that are vertical or horizontal, there are specific forms.

• If a parabola opens up or down, its standard form is \( (x - h)^2 = 4p(y - k) \).
• If it opens to the left or right, like in our problem, the form is \( (y - k)^2 = 4p(x - h) \).

In our exercise:

• The derived equation is \( y^2 = -40x \), indicating a horizontal parabola that opens left, since 4p is negative.

This form is exceedingly useful as it directly exhibits the vertex and the parameter \(p\), which assists in revealing the focus or directrix alignment.

Horizontal Parabola

A horizontal parabola, unlike its more commonly discussed vertical counterpart, opens side-to-side rather than up or down. In the given exercise, we encounter such a parabola.
Being familiar with its traits is crucial for recognizing and solving problems involving horizontal parabolas.

• The orientation stems from its focus and directrix configuration, alongside the calculation of \(p\), the distance parameter.

Typically:

• If \(p\) is positive, the parabola opens to the right.
• If \(p\) is negative, like ours, it opens to the left.

The equation \(y^2 = -40x\) reveals that our parabola is horizontal, shaped by negation in its \(4p\) component, and emphasized by its opening direction along the x-axis. This understanding aids in navigating problems where the parabola's non-vertical orientation plays a central role.