Question

Find the equation of the parabola defined by the given information. Sketch the parabola. Focus: (3,2)\(;\) directrix: \(y=1\)

Short answer

The equation of the parabola is \(y = \frac{1}{2}x^2 - 3x + 6\), opening upward with vertex (3, 1.5).

Step-by-step solution

Understand the Geometry of the Parabola

A parabola is defined as the set of all points that are equidistant from a point called the "focus" and a line called the "directrix." Here, the focus is at (3,2) and the directrix is the line \(y = 1\).

Set Up the Distance Formula

Use the definition of a parabola to set up an equation where any point \((x, y)\) on the parabola is equidistant to the focus \((3, 2)\) and the line \(y = 1\). The distance from a point \((x, y)\) to the focus is \(\sqrt{(x - 3)^2 + (y - 2)^2}\). The distance from a point \((x, y)\) to the directrix \(y = 1\) is \(|y - 1|\).

Equate the Distances

Set the distance from the point \((x, y)\) to the focus equal to the distance from the point \((x, y)\) to the directrix. This gives us the equation:\[ \sqrt{(x - 3)^2 + (y - 2)^2} = |y - 1| \]

Simplify the Equation

Square both sides of the equation to remove the square root.\[(x - 3)^2 + (y - 2)^2 = (y - 1)^2\]Expand and simplify:\[x^2 - 6x + 9 + y^2 - 4y + 4 = y^2 - 2y + 1\]

Rearrange to Find the Parabola Equation

Remove \(y^2\) and combine like terms:\[x^2 - 6x + 9 - 4y + 4 = -2y + 1\]Simplifying further:\[x^2 - 6x + 12 = 2y\]And then rearrange it to the standard form of the parabola equation:\[y = \frac{1}{2}x^2 - 3x + 6\]

Sketch the Parabola

Plot the focus (3,2) and the directrix \(y = 1\) on a coordinate plane. Then, using the vertex form \(y = \frac{1}{2}(x - 3)^2 + 2\) derived from the standard form, sketch the parabola, noting that it opens upward with its vertex at the midpoint between the focus and the directrix at \((3, 1.5)\).

Key concepts

Focus

The focus of a parabola is a special point that helps in shaping the curve. For any point on the parabola, the distance to the focus is equal to the distance to the directrix, which is a straight line. In this exercise, the focus is located at the coordinates (3,2), which serves as a central reference point for the shape of the parabola. This creates symmetry and defines how the curve opens. When sketching or defining the parabola, visualize the focus as a beacon, pulling the curve towards it. The presence of the focus ensures that the parabola is not just any curve, but one with a significant physical property that balances distances across its structure. Understanding this is crucial as it lays the groundwork for constructing and analyzing parabolas.

Directrix

The directrix is a line, typically horizontal or vertical, that works alongside the focus to define a parabola. It's as pivotal as the focus, creating a balance within the structure of the curve. In this problem, the directrix is given as the line \(y = 1\). This line acts as an invisible barrier that helps guide the parabola's arc seamlessly.The parabola curves in such a way that it maintains constant distance to both the focus and the directrix. This symmetrical property means that for any point on the parabola, how far it is from the focus is exactly mirrored in how far it is from the directrix. Therefore, the directrix serves as an essential concept to grasp when delving into the geometric definitions of conic sections.

Distance Formula

The distance formula is a mathematical tool used to determine the distance between two points in a plane. In the context of parabolas, this formula defines the relationship between any point on the curve, the focus, and the directrix. It ensures that each point on the curve is equidistant to the focus and the directrix, maintaining the foundational property of a parabola.To apply the distance formula in this exercise, we calculate two distances:

• From any point \((x, y)\) on the parabola to the focus \((3, 2)\), represented by \(\sqrt{(x - 3)^2 + (y - 2)^2}\).
• From the same point \((x, y)\) to the directrix \(y = 1\), expressed as \(|y - 1|\).

By setting these distances equal, we obtain the equation of the parabola. This critical step translates the geometric definition into a solvable algebraic form, leading to practical applications in solving and graphing.

Vertex Form

The vertex form of a parabola offers a straightforward way to identify its vertex, the highest or lowest point on the curve depending on its orientation. For the parabola given in the exercise, the transformation to vertex form is achieved by manipulating the standard form equation \(y = \frac{1}{2}x^2 - 3x + 6\).After completing the square and other algebraic steps, the equation can be rewritten in the vertex form:\[y = \frac{1}{2}(x - 3)^2 + 2\]This reveals that the vertex is located at (3, 1.5), which is exactly between the focus and the directrix. Understanding the vertex form is beneficial as it provides insights directly into the geometry of the parabola, such as the direction it opens and its steepness. It also offers a convenient method to sketch the curve, emphasizing the symmetry centered at the vertex.