Problem 40 - Precalculus: Mathematics for Calculus

Question

Find an equation for the conic section with the given properties. The parabola with focus \(F(1,3)\) and directrix \(x=3\)

Step-by-Step Solution

Answer

The equation is \((y - 3)^2 = -4(x - 2).\)

Step 1: Identify the Vertex

The vertex of a parabola is located halfway between the focus and the directrix. Given the focus is \( F(1, 3) \) and the directrix is \( x = 3 \), the vertex, \( V \), is at \( x = 2 \) (midpoint of \( x = 1 \) and \( x = 3 \)) and the same \( y \)-coordinate as the focus, \( y = 3 \). Thus, the vertex is \( V(2, 3) \).

Step 2: Determine the Orientation

The directrix is a vertical line, \( x = 3 \), indicating that the parabola opens horizontally. Since the focus is to the left of the directrix, the parabola opens to the left.

Step 3: Find the Equation Using the Parabola Formula

For a parabola opening horizontally with vertex \( (h, k) \) at \( (2, 3) \), the equation is \[(y - k)^2 = 4p(x - h),\]where \( p \) is the distance from the vertex to the focus. The distance \( p \) between \( x = 2 \) and \( x = 1 \) is \( p = -1 \), since the direction is to the left. Hence, the equation is \[(y - 3)^2 = -4(x - 2).\]

Key Concepts

Focus and Directrix

The focus and directrix are fundamental in understanding parabolas. Imagine a point, called the focus, and a line, called the directrix. A parabola is a curve where every point is equidistant from the focus and directrix.
This characteristic is key in deriving the parabola's properties. In our example, the focus is at \( F(1, 3) \) and the directrix is the line \( x = 3 \).
To find the position of the parabola, we use the relationship between these two components. Each point on the parabola maintains equilibrium, maintaining equal distances to both the focus and the directrix. This property helps us in construction and equation formulation of parabolas.

Vertex

The vertex is the turning point of a parabola and is located midway between the focus and the directrix. In simpler terms, it's the point where the parabola changes direction.
To find the vertex with given focus and directrix, calculate the midpoint along the horizontal or vertical axis (depending on orientation) between the focus and directrix.
This makes it easier to understand why the vertex for the parabola given by the focus \( F(1,3) \) and directrix \( x=3 \) is at \( V(2,3) \). It’s right in the middle on the horizontal axis between the focus and directrix. The \( y \)-coordinate of the vertex remains the same as the focus since they align horizontally.

Parabola Equation

To express a parabola mathematically, we employ a specific formula. For parabolas that open horizontally, the equation resembles \((y - k)^2 = 4p(x - h)\).
This formula arises from the relationship of a parabola's geometric properties. Here, \((h, k)\) represents the vertex. The value \( p \) indicates how far the focus is from the vertex.
In the exercise, we determine \( p = -1 \) because the parabola opens towards the left from the vertex \((2,3)\). The negative sign indicates leftward direction. Consequently, the equation for the given parabola is \((y - 3)^2 = -4(x - 2)\). By mastering this formula, you can easily draft different parabola equations based on their focus, directrix, and orientation.

Conic Sections

Conic sections are curves derived from the intersection of a plane and a cone. They include ellipses, circles, parabolas, and hyperbolas.
Parabolas are just one type of conic section, originating from parallel cuts to the sides of the cone. Each conic section has unique properties and equations.
This concept helps to visualize different curves we observe in math and nature. Conic sections all follow certain mathematical principles that help us understand their formation and behavior.

• Ellipses are closed loops.
• Circles are a special form of ellipses where both axes are equal.
• Hyperbolas consist of two disconnected curves.
• Parabolas, like in our exercise, are open curves.

Understanding these concepts provides insight into geometric relationships and enhances problem-solving skills.