Problem 34 - Contemporary Precalculus

Question

Find the polar equation of the conic section that has focus (0,0) and satisfies the given conditions. Parabola; vertex \((2, \pi / 2)\)

Step-by-Step Solution

Answer

The polar equation of the parabola is: \(r^2\cos^2\theta = 8r^3(1-\cos^2\theta) - 16r^2\)

Step 1: Identify the conic section and set up the equation

The conic section in question is a parabola, with vertex at \((2, \pi/2)\) and focus at the origin, \((0,0)\). To find the polar equation of this parabola, we need to express the equation with polar coordinates (r, θ) instead of Cartesian coordinates (x, y).

Step 2: Convert the vertex from polar to Cartesian coordinates

The polar coordinates of the vertex are given by \((2, \pi/2)\). To find the equivalent Cartesian coordinates, we use the following conversion formula: \(x = r\cos\theta\) \(y = r\sin\theta\) So, for the vertex \((2, \pi/2)\): \(x = 2\cos(\pi/2) = 0\) \(y = 2\sin(\pi/2) = 2\) Therefore, the vertex of the parabola in Cartesian coordinates is \((0,2)\).

Step 3: Write the equation of a parabola with vertex (0,2) and focus (0,0)

We know that the standard equation of the parabola, facing downwards with vertex at \((0,2)\) and focus \((0,0)\), is: \((x-h)^2 = 4p(y-k)\) Where (h,k) is the vertex and p is the distance from the vertex to the focus. Since the vertex is \((0,2)\) and the focus is the origin, we have \((h,k)=(0,2)\) and \(p = 2\). So, the equation of the parabola in Cartesian coordinates is: \(x^2 = 4(2)(y-2)\) \(x^2 = 8(y-2)\)

Step 4: Convert the Cartesian equation to a polar equation

Start by substituting the polar-to-Cartesian conversion formulas from Step 2: \(x = r\cos\theta\) \(y = r\sin\theta\) Into the Cartesian equation: \((r\cos\theta)^2 = 8(r\sin\theta - 2)\)

Step 5: Simplify the polar equation

To simplify the polar equation, we use the following trigonometric identity: \(\sin^2\theta + \cos^2\theta = 1\) Multiply our polar equation by \(r^2\): \(r^2\cos^2\theta = 8r^3\sin\theta - 16r^2\) Now, we can substitute \(\sin^2\theta = 1 - \cos^2\theta\): \(r^2\cos^2\theta = 8r^3(1-\cos^2\theta) - 16r^2\) This equation gives us the polar equation of the parabola with focus (0,0) and vertex (2, π/2).

Key Concepts

Parabola in Polar Coordinates

Understanding the shape and equation of a parabola in polar coordinates can be challenging, but it's essential for visualizing and solving problems related to conics in a polar context. A parabola is one of the conic sections which can be defined as the set of all points in a plane that are equidistant from a fixed point, called the focus, and a fixed line known as the directrix.

When the focus is at the origin (0,0) and the directrix is perpendicular to the polar axis, the parabola's equation in polar coordinates can be generally written as:\[ r = \frac{e}{1 + e\cos\theta} \]where \(e\) is the eccentricity of the conic, which is 1 for a parabola. The variable \(r\) is the distance from the origin to a point on the curve, and \(\theta\) is the angle formed by the positive x-axis and the line segment joining the origin to the point on the curve. Understanding this relationship helps students visualize how the curves of parabolas are plotted in a polar coordinate system and how they differ from other conics.

Converting Cartesian to Polar Coordinates

Transitioning between Cartesian and polar coordinates is a common exercise in mathematics, particularly when dealing with conic sections. Cartesian coordinates express each point through an ordered pair \((x, y)\), while polar coordinates use \((r, \theta)\), where \(r\) is the radius or the distance from the origin to the point, and \(\theta\) is the angle from the positive x-axis to the line segment connecting the origin to the point.

The conversion from Cartesian to polar coordinates is given by:\[ r = \sqrt{x^2 + y^2} \]\[ \theta = \arctan\frac{y}{x} \]Similarly, to convert from polar back to Cartesian coordinates, use:\[ x = r\cos\theta \]\[ y = r\sin\theta \]These transformations are pivotal as they allow us to rewrite equations and interpret geometric figures in the most convenient coordinate system for the problem at hand, thereby simplifying the visualization and solution process.

Vertex and Focus of a Parabola

The vertex and focus are significant components of a parabola's definition and its equation. The vertex is the point where the curvature is most pronounced, and the graph changes direction. In the case of a parabola in polar coordinates with the focus at the origin, the vertex is the closest point on the graph to the focus.

The focus, on the other hand, is a fixed point that, in combination with the directrix, defines a conic section. For a parabola, the distance from any point on the curve to the focus is equal to the perpendicular distance to the directrix. This property is harnessed to derive the equation of a parabola.

Knowing the coordinates of the vertex and focus allows us to write the conic's equation more easily either in Cartesian or polar form. For instance, if a parabola has a vertex at \((h, k)\) in Cartesian coordinates and its focus is at the origin, then the parabola's equation can be expressed as \((x-h)^2 = 4p(y-k)\) where \(p\) represents the distance from the vertex to the focus. These elements are not just abstract notions but serve as the foundational components from which the entire structure of the curve emanates.