Problem 3 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the coordinates of the focus and the equation of the directrix for each parabola. Make a sketch showing the parabola, its focus, and its directrix. $$ x^{2}=-12 y $$

Step-by-Step Solution

Answer

The focus is at \((0, -3)\) and the directrix is \(y = 3\).

Step 1: Identify the Parabola Form

The given equation is \(x^2 = -12y\). This equation is of the form \(x^2 = 4py\), which represents a parabola that opens downwards if \(p < 0\) (since the coefficient of \(y\) is negative).

Step 2: Determine the Value of p

To find the value of \(p\), we compare \(x^2 = 4py\) with \(x^2 = -12y\). Here, \(4p = -12\). Solve for \(p\) by dividing both sides of the equation by 4 to get \(p = -3\). This means the parabola opens downwards with vertex \((0,0)\).

Step 3: Find the Focus of the Parabola

The vertex of this parabola is at the origin \((0,0)\). The focus of a parabola \(x^2 = 4py\) is at \( (0, p)\). Substitute \(p = -3\) to get the focus \((0, -3)\).

Step 4: Determine the Directrix

The directrix of a parabola \(x^2 = 4py\) is a horizontal line \(y = -p\). Substitute \(p = -3\) to find the directrix: \(y = 3\).

Step 5: Sketch the Parabola with Focus and Directrix

Draw the vertical \(y\)-axis and horizontal \(x\)-axis. The vertex of your parabola is \((0,0)\). Plot the focus at \((0, -3)\) and draw a horizontal line for the directrix at \(y = 3\). Sketch the parabola opening downward to indicate its direction based on the focus and directrix.

Key Concepts

Focus of a Parabola

The focus of a parabola is a special point that helps define the curve. When dealing with a parabola, it is essential to know where the focus is located.
The focus is the point from which all points on the parabola are equidistant to a line called the directrix and the focus itself.
In the equation of a parabola given as \( x^2 = 4py \), the focus is located at \( (0, p) \). For the example equation \( x^2 = -12y \), we determined that \( p = -3 \), placing the focus at the coordinates\( (0, -3) \). This specific location of the focus will determine whether the parabola opens upward or downward.
The closer the focus is to the vertex, the 'sharper' or 'narrower' the parabola will appear. Understanding the position of the focus is crucial for graphing the parabola correctly.

Directrix of a Parabola

The directrix of a parabola is a significant element, being a line that is perpendicular to the axis of symmetry of the parabola.
In simple terms, the directrix works alongside the focus to define the shape of the parabola.
The distance from any point on the parabola to the focus is equal to the perpendicular distance from that point to the directrix.
In the standard form \( x^2 = 4py \), the directrix is given by the equation \( y = -p \). For our example equation \( x^2 = -12y \), substituting \( p = -3 \) results in the directrix being \( y = 3 \).
This implies a horizontal line at \( y = 3 \) in a graph. The concept of the directrix is fundamental in understanding how parabolas are shaped and how they relate geometrically to other elements like the focus.

Equation of a Parabola

The equation of a parabola dictates its orientation and shape.
The basic form of a parabolic equation is \( x^2 = 4py \), which shows a parabola with its vertex at the origin \((0,0)\) and axis of symmetry along the y-axis.
In this scenario, if \( p \) is positive, the parabola opens upwards, and if \( p \) is negative, as in \( x^2 = -12y \), it opens downwards.
This fundamental structure transforms into other various orientations and translations, but the relationship with \( p \) remains key.
Factors such as concavity (the direction in which it opens) and the openness (determined by the value of \( p \)) are all dependent on this simple, yet versatile form.
Understanding this form enables you to determine the positioning of other elements such as the focus and directrix efficiently.

Conic Sections

Parabolas are part of a fascinating family of shapes called conic sections.
Conic sections arise from the intersection of a plane with a cone, producing various shapes like circles, ellipses, parabolas, and hyperbolas.
A parabola specifically is formed when the plane is parallel to the slope of the cone.
Each type of conic section has distinct characteristics:

• Circle: Perfectly round with a center and radius.
• Ellipse: An elongated circle that has two foci.
• Parabola: A U-shaped curve discussed in detail here.
• Hyperbola: Two open curves opposite each other.

While parabolas might seem just like any curvy line, their importance extends to fields like physics and engineering due to their reflective properties and behavior under quadratic equations. Embracing their place in conic sections provides deeper insight into both mathematics and its applications.