Problem 54 - Precalculus

Question

Convert the equation from polar to rectangular form. Identify the resulting equation as a line, parabola, or circle. $$r^{2} \cos ^{2} \theta-r \sin \theta=-2$$

Step-by-Step Solution

Answer

The rectangular form is \(x^2 - y + 2 = 0\), which represents a parabola.

Step 1: Understand the Problem

We need to convert the polar equation \(r^{2} \cos^{2} \theta - r \sin \theta = -2\) into rectangular form. In polar coordinates, \(r\) is the radial distance from the origin, and \(\theta\) is the angle. In rectangular coordinates, we use \(x\) and \(y\).

Step 2: Use Polar-Rectangular Conversion Formulas

The conversion between polar and rectangular coordinates involves the following formulas: \(x = r \cos \theta\), \(y = r \sin \theta\), and \(r^2 = x^2 + y^2\). We will use these to convert the given equation.

Step 3: Substitute for \(r\) and \(\theta\)

Substitute \(r \cos \theta\) with \(x\) and \(r \sin \theta\) with \(y\). First, recognize that \(\cos^2 \theta = (\frac{x}{r})^2 = \frac{x^2}{r^2}\), and substitute appropriately:\[\frac{x^2}{r^2} \cdot r^2 - r \sin \theta = -2\]

Step 4: Simplify the Equation

Cancel \(r^2\) from the first term:\[x^2 - y = -2\]This is now in a simplified rectangular form. Rearrange to identify the form:\[x^2 - y + 2 = 0\]

Step 5: Identify the Type of Curve

Observe the simplified equation \(x^2 - y + 2 = 0\). This equation can be rewritten as \(x^2 = y - 2\), which is a standard form of a parabola that opens upwards. The focus of the parabola is vertically above the vertex.

Key Concepts

Polar Coordinates

Polar coordinates provide a unique way of representing points in a plane. Unlike the traditional rectangular coordinates, where every point is described by an x-coordinate and a y-coordinate, polar coordinates use a different set of information. In the polar system, a point is identified by the radial distance from the origin, denoted as \( r \), and the angle, \( \theta \), that the radial line makes with the positive x-axis.

Key features of polar coordinates include:

• The radial distance \( r \), which tells you how far the point is from the origin. \( r \) is positive for points in the direction of \( \theta \), and negative for points in the opposite direction.
• The angle \( \theta \), measured in radians, helps in identifying the direction of the point from the origin.

Converting between polar and rectangular coordinates can provide new insights or simplify solving problems, depending on the type of symmetry or orientation of the figure in question.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a way to pinpoint the exact location of a point in a two-dimensional plane using two numbers: \( x \) and \( y \). These coordinates are based on a grid-like system with two perpendicular axes, the x-axis, and the y-axis.

Important aspects of rectangular coordinates:

• The \( x \)-coordinate indicates the horizontal distance of the point from the vertical y-axis.
• The \( y \)-coordinate indicates the vertical distance of the point from the horizontal x-axis.

While polar coordinates focus on direction and distance, rectangular coordinates provide a way to easily perform algebraic and geometric calculations. Transforming polar equations into rectangular form can be useful in simplifying and solving equations involving lines and curves in standard forms such as circles, ellipses, and parabolas.

Parabola

A parabola is a symmetrical plane curve that is an important concept in mathematics, especially in the study of quadratic functions and conic sections. It is the set of all points equidistant from a fixed point (the focus) and a fixed straight line (the directrix).

Characteristics of a parabola:

• The vertex is the point where the curve changes direction and is the midpoint between the focus and the directrix.
• The axis of symmetry is a vertical or horizontal line that passes through the vertex, dividing the parabola into two mirror-image halves.
• The parabola can open upwards, downwards, left, or right, depending on its equation.

The equation derived from polar to rectangular conversion, \( x^2 = y - 2 \), is in the standard form of a vertical parabola that opens upwards. Understanding this form helps in identifying key features like the vertex at \( (0,2) \) and the shape of the graph. Recognizing such equations allows for easier graphing and further analysis using algebraic methods.