Problem 33 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 33-34: Find a polar equation of the parabola with focus at the pole and the given vertex. $$ V\left(4, \frac{\pi}{2}\right) $$

Step-by-Step Solution

Answer

The polar equation is \(r = \frac{4}{1 + \sin(\theta)}\).

Step 1: Understand the Problem

We need to find a polar equation for a parabola with its focus at the pole (origin), and the vertex is given at polar coordinates \((4, \frac{\pi}{2})\). This means the vertex is located 4 units away at an angle of \(\frac{\pi}{2}\) radians from the positive x-axis.

Step 2: Identify the Parabola's Characteristics

In polar coordinates, a parabola centered with focus at the pole has the form \(r = \frac{ed}{1 + e\cos(\theta - \phi)}\), where \(e = 1\) for a parabola. \(d\) is the distance from the focus (at pole) to the vertex, and \(\phi\) is the angle from the positive x-axis to the direction of the vertex.

Step 3: Write the General Polar Equation for the Parabola

Since the vertex is at \((4, \frac{\pi}{2})\), we know \(d = 4\) and \(\phi = \frac{\pi}{2}\). Substitute \(e = 1\), \(d = 4\), and \(\phi = \frac{\pi}{2}\) into the equation to get:\[r = \frac{4}{1 + \sin(\theta)}\] because \(\cos(\theta - \frac{\pi}{2}) = \sin(\theta)\).

Step 4: Finalize the Equation

The polar equation of the parabola can be simplified and verified as: \[r = \frac{4}{1 + \sin(\theta)}\].This is the equation of the parabola with its focus at the pole, and the vertex at \((4, \frac{\pi}{2})\).

Key Concepts

Parabola

The parabola is a unique conic section, a curve where any given point is at an equal distance from a fixed point called the "focus" and a line known as the "directrix." In the context of polar equations, the parabola is represented with its focus at the pole, or the origin point of the coordinate system. This makes it a bit more straightforward when working with curves in a radial manner rather than the classic Cartesian plane for a parabola like \(y = ax^2 + bx + c\). The defining feature of a parabola is its eccentricity \(e = 1\), which differentiates it from other conical curves like ellipses and hyperbolas. The role of the focus and the alignment of the vertex are crucial in determining the exact shape and position of the parabola in the polar coordinate system.

Focus and Vertex

In any parabola, the focus and vertex are key elements that help describe its geometry. The focus of a parabola is a fixed point that, together with the directrix, helps in the trajectory definition of the curve. For polar equations, the focus is located at the pole, making the calculus simpler for certain applications. The vertex, alternatively, is the point on the parabola that is closest to the focus.
In this problem, the vertex is particularly salient because it is defined using polar coordinates:

• Vertex: \((4, \frac{\pi}{2})\), meaning it is 4 units from the origin and oriented at an angle of \(\frac{\pi}{2}\) radians.

With the vertex and focus known, one can derive the polar equation necessary to describe the overall shape of the parabola.

Polar Coordinates

Polar coordinates represent a two-dimensional coordinate system where each point on a plane is determined by an angle and a distance from a fixed center point. They are especially useful in dealing with curves that radiate from a central point, such as circles or parabolas. The coordinates are usually expressed in the form \((r, \theta)\), where \(r\) is the radius or distance from the pole, and \(\theta\) is the angle in radians.
In problems involving polar coordinates, like the example you are studying, the focus (located at the pole) and the orientation of the vertex play important roles in determining the equation of a curve. Polar coordinates simplify many mathematical tasks involving rays and angles, allowing easier manipulation of equations concerning conic sections.

Radian Measure

Radian measure is a way of measuring angles based on the length of the arc created by the angle on the unit circle. One radian is the angle created when the arc length is equal to the radius of the circle. While degrees are more commonly used in everyday measurement within geographic navigation or astronomic calculations, radians are preferred in most mathematical contexts because they offer cleaner integration and differentiation results.
The relationship between degrees and radians is given by:

• \(180^{\circ} = \pi\) radians
• \(360^{\circ} = 2\pi\) radians

For this exercise, when looking at the vertex \((4, \frac{\pi}{2})\), \(\frac{\pi}{2}\) corresponds to \(90^{\circ}\), indicating a direction perpendicular to the positive x-axis. Gaining familiarity with radians helps in understanding trigonometric functions and when working with polar equations, especially when calculating coordinates or angles of a parabola.