Question

Use the Distance Formula to derive the equation of a parabola that opens to the right with vertex \((0,0)\), focus \((p, 0)\), and directrix \(x=-p\).

Short answer

The equation of the parabola that opens to the right with vertex \((0,0)\), focus \((p, 0)\), and directrix \(x=-p\) is \(x^2 = 4py\).

Step-by-step solution

Understanding the definitions

By definition, a parabola is the loci of points that are equidistant from a fixed line (the directrix) and a fixed point (the focus). In this case, the focus is \((p, 0)\) and the directrix is \(x=-p\). Hence, for a point \((x,y)\) on the parabola, the distance to the focus should be equal to the distance to the directrix.

Setup distance formula

Use the distance formula to express the distances mentioned in Step 1. The distance from a point \((x,y)\) to the focus \((p, 0)\) is \(\sqrt{(p-x)^2 + y^2}\). The distance to the directrix \(x=-p\) is \(x--p = x+p\). Since the point on the parabola is equidistant to the focus and the directrix, we set these two expressions equal to each other: \(\sqrt{(p-x)^2 +y^2}= x+p\).

Solve the equation

Solve the equation obtained in Step 2 to get the equation of the parabola. Start by squaring both sides of the equation to get rid of the square root: \( (p-x)^2+y^2 = (x+p)^2\). Expand and simplify this to get the equation: \(x^2 = 4py\), where \(x\) is the horizontal distance and \(y\) is the vertical distance from the vertex to the point on the parabola. This equation indicates that the parabola opens to the right since \(x\) is positive when the parabola opens to the right.

Key concepts

Parabola

A parabola is one of the key shapes in mathematics, especially under the study of conic sections. You can visualize it as a U-shaped curve. What makes a parabola unique is its geometric property that every point on the parabola is equidistant from a single point called the focus and a line called the directrix. You'll often see parabolas in physics and engineering due to their reflective properties.

Parabolas have different orientations based on how they are defined. They can open either up, down, left, or right. For example, if you think of a satellite dish, it is essentially a parabolic shape that reflects signals to a receiver located at the focus. The direction the parabola opens is determined by its equation.

Vertex

The vertex is a defining point of the parabola. It represents the point of turn, or the maximum or minimum point, depending on the direction of the parabola. It's essentially at the heart of the parabola. In your exercise, the vertex is (0, 0), which means the parabola is centered at the origin of the coordinate system.

The location of the vertex affects the symmetry of the parabola. When you have equations like \(x^2 = 4py\), the vertex is in a standard position and relates directly to the axis of the parabola's symmetry. This makes it easier during calculations and graphing.

Focus

The focus of a parabola is crucial. It's that fixed point that every point on the parabola is equidistant to its matching point on the directrix. In your exercise, the focus is \((p, 0)\). This tells us that the parabola opens horizontally towards the right side. The focus helps determine where the parabola opens and its orientation.

Imagine a flashlight. The bulb is at the focus, and the light rays reflect off the parabola to form a beam, following this unique property of parabolas. Knowing where the focus is leads to understanding the dimensions and scale of the parabolic curve, as the distance between the vertex and focus reflects how "wide" or "narrow" the parabola might appear.

Directrix

The directrix is like the polar opposite of the focus. It is a straight line that is also equidistant from any point on the parabola with respect to the focus. In this exercise, the directrix is represented by the vertical line \(x = -p\).

While the focus deals with a point, the directrix provides a line as a reference. Both of these influence the curvature and positioning of the parabola. When you're trying to write the equation of a parabola, you'll use the distance to the directrix as part of your calculations. In physics, the concept of the directrix is used in satellite dishes and telescopes, where precise focus is needed to process or collect signals efficiently.