Problem 9 - Thomas Calculus

Question

Exercises \(9-16\) give equations of parabolas. Find each parabola's focus and directrix. Then sketch the parabola. Include the focus and directrix in your sketch. $$ y^{2}=12 x $$

Step-by-Step Solution

Answer

Focus: \((3, 0)\), Directrix: \(x = -3\)

Step 1: Identify the Equation Type

Recognize that the given equation \( y^2 = 12x \) represents a parabola. Since \( y \) is squared, this is a horizontal parabola that opens to the right.

Step 2: Convert to Standard Form

The standard form of a horizontal parabola is \( y^2 = 4px \). In the equation \( y^2 = 12x \), compare to the standard form to find \( 4p = 12 \). Solving for \( p \) gives \( p = 3 \).

Step 3: Calculate the Focus

For the parabola \( y^2 = 4px \), the focus is located at \((p, 0)\). Therefore, with \( p = 3 \), the focus is at \((3, 0)\).

Step 4: Determine the Directrix

For the parabola \( y^2 = 4px \), the directrix has the equation \( x = -p \). Thus, the directrix of the given parabola is \( x = -3 \).

Step 5: Sketch the Parabola

Draw the parabola that opens to the right with the vertex at \( (0,0) \). Plot the focus at \( (3, 0) \) and draw the directrix line at \( x = -3 \). Make sure the parabola is equidistant from the focus and the directrix at any point on the curve.

Key Concepts

Focus of a Parabola

The focus of a parabola is a special point that plays a crucial role in defining the parabola's shape and position. When dealing with a horizontal parabola like the one given by the equation \( y^2 = 4px \), the focus is positioned at \((p, 0)\). Here, \( p \) is a constant that determines how "stretched" the parabola is along its directrix.To find the focus, compare the equation of your parabola to the standard form. For example, in the exercise \( y^2 = 12x \), notice the similarity with \( y^2 = 4px \). By matching these equations, you'll find that \( 4p = 12 \), giving us \( p = 3 \). Thus, the focus for this parabola is \((3, 0)\). Remember, the focus is crucial for understanding the parabola's reflective properties. Every point on the curve is equidistant from both the focus and the directrix, helping objects follow a smooth path along the parabola.

Directrix of a Parabola

The directrix of a parabola is a fixed line that, together with the focus, helps to define the parabola. For a horizontal parabola like \( y^2 = 4px \), the directrix runs parallel to the \( y \)-axis. Its role is to ensure that each point on the parabola is at an equal distance from both the focus and the directrix line.In our example, with the equation \( y^2 = 12x \), we've previously determined that \( p = 3 \). The formula for the directrix of a horizontal parabola is \( x = -p \). Hence, for our parabola, the directrix is the vertical line \( x = -3 \).The directrix helps guide the shape of the parabola. By imagining it as a sort of invisible mirror, you can better understand how light or sound waves might reflect off the parabolic surface, determining pathways that maintain equidistance to both focus and directrix.

Horizontal Parabolas

Horizontal parabolas have the form \( y^2 = 4px \), which distinguishes them from the more familiar vertical parabolas \( x^2 = 4py \). In horizontal parabolas, the square is on the \( y \) variable, meaning the parabola opens sideways, rather than up or down. This orientation can be crucial for applications like satellite dishes where side-focused reception is needed.In a horizontal parabola, the vertex is typically located at the origin \( (0,0) \) unless otherwise translated. The graph of the parabola will open either towards the positive \( x \)-axis or the negative \( x \)-axis, depending on the value of \( p \). For \( p > 0 \), the parabola opens to the right, as in our equation \( y^2 = 12x \), and for \( p < 0 \), it opens to the left.These parabolas have unique attributes, such as having their directrix parallel to the \( y \)-axis, and this alignment aids in tasks that require specific directional focal capabilities.