Problem 3 - Algebra and Trigonometry

Question

The graph of the equation \(y^{2}=4 p x\) is a parabola with focus F (_____,_____) and directrix \(x=\) _____ So the graph of \(y^{2}=12 x\) is a parabola with focus \(F\) (_____,_____) and directrix \(x=\) _____.

Step-by-Step Solution

Answer

Focus: (3, 0); Directrix: x = -3.

Step 1: Identify the Given Equation Form

The equation of the form \( y^2 = 4px \) represents a parabola with its vertex at the origin \( (0, 0) \). We compare the given equation \( y^2 = 12x \) with this standard form.

Step 2: Determine the Value of p

From the equation \( y^2 = 4px \), the term \( 4p \) is equal to 12 in the equation \( y^2 = 12x \). Solving for \( p \), we find: \( 4p = 12 \) \( \Rightarrow p = \frac{12}{4} = 3 \).

Step 3: Find the Focus

The focus of a parabola given by \( y^2 = 4px \) is \( (p, 0) \). Here, \( p = 3 \), so the focus is \( (3, 0) \).

Step 4: Determine the Directrix

The directrix of the parabola \( y^2 = 4px \) is \( x = -p \). Substituting \( p = 3 \), we find the directrix is \( x = -3 \).

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a crucial point as it represents the point where the parabola is at its most curved. It acts like the "tip" of the curve. The standard parabola, defined by the equation \(y^2 = 4px\), has its vertex at the origin \((0,0)\) when centered. This means that all measurements and other features, like the focus and directrix, are defined in relation to this point. For parabolas that are not centered at the origin, other forms of the equation may be needed to identify the vertex, such as \((h,k)\) in a translated parabola equation.

Standard Form of a Parabola Equation

The standard form of a parabolic equation can vary depending on the orientation of the parabola. For a parabola that opens sideways, the form is typically \(y^2 = 4px\). This shows that the parabola opens either to the right or left along the x-axis. In this equation, \(4p\) is a coefficient that helps in defining the position of the focus and the directrix.

When you compare any given parabola equation to this standard form, it becomes easier to derive other properties of the parabola, like the focus and the directrix. The importance of the number 4 in the equation \(4p\) can't be understated, as it standardizes the equation, making it easier to understand the parabola's fundamental components.

Solving for Parameter p

The parameter \(p\) is essential for determining the parabola's geometry. It pertains to the horizontal displacement for parabolas opening sideways. In the equation \(y^2 = 4px\), \(p\) is directly responsible for determining the distance from the vertex to the focus and from the vertex to the directrix.

To solve for \(p\), consider the equation provided, \(y^2 = 12x\). Here, the comparison to the standard form \(y^2 = 4px\) shows us that \(4p = 12\). By solving this equation for \(p\), we divide both sides by 4, resulting in \(p = 3\). This means that both the focus and directrix are located at a distance of 3 units from the vertex along the x-axis, with the focus at \((3, 0)\) and the directrix at \(x = -3\). Understanding \(p\) aids not only in graphing the parabola correctly but also in recognizing its symmetry and direction.