Problem 56 - College Algebra

Question

Recall from Rotation of Axes that equations of conics with an \(x y\) term have rotated graphs. For the following exercises, express each equation in polar form with \(r\) as a function of \(\theta\). $$ x y=2 $$

Step-by-Step Solution

Answer

The polar form is \( r = \pm \frac{2}{\sqrt{\sin(2\theta)}} \).

Step 1: Identify the relationship between rectangular and polar coordinates

In rectangular coordinates, we have \(x\) and \(y\) as the variables. To convert this into polar form, we use the relationships \(x = r\cos\theta\) and \(y = r\sin\theta\).

Step 2: Substitute polar coordinate expressions into the equation

Replace \(x\) and \(y\) in the equation \(xy = 2\) with the expressions in terms of \(r\) and \(\theta\). This gives us \((r\cos\theta)(r\sin\theta) = 2\).

Step 3: Simplify the equation

In the equation \((r\cos\theta)(r\sin\theta) = 2\), combine the terms on the left to obtain \(r^2 \cos\theta \sin\theta = 2\).

Step 4: Solve the equation for \(r\) as a function of \(\theta\)

Since we need to express this equation as \(r\) as a function of \(\theta\), divide both sides by \(\cos\theta \sin\theta\): \[ r^2 = \frac{2}{\cos \theta \sin \theta} \]. Based on the identity \(\cos \theta \sin \theta = \frac{1}{2}\sin(2\theta)\), solve for \(r^2\) to get \[ r^2 = \frac{4}{\sin(2\theta)} \]. Finally, solve for \(r\): \[ r = \pm\frac{2}{\sqrt{\sin(2\theta)}} \].

Key Concepts

Conics

Conic sections, or simply conics, are curves formed by the intersection of a plane with a double-napped cone. Depending on the angle and location of the intersection, the conic section might be a circle, an ellipse, a parabola, or a hyperbola. These curves are important in various scientific fields, such as physics, engineering, and astronomy, due to their geometric properties.

The standard form for each varies:

• A circle: all points equidistant from a center point
• An ellipse: a stretched circle with two focal points
• A parabola: points equidistant from a focus and a directrix
• A hyperbola: two disparate curves mirroring each other's placement

Understanding conics in different coordinate systems opens up more analysis and application opportunities. Hence, transforming their equations into polar coordinates forms part of this understanding.

Rotation of Axes

The rotation of axes is a technique used to eliminate the xy-term in the equation of a conic, by rotating the axes to align them with the conic's principal axes. This simplifies the equation significantly.

When we rotate by an angle \(\theta\), the new coordinates \((x', y')\) relate to the old \((x, y)\) through the transformation equations:

• \(x = x' \cos \theta - y' \sin \theta\)
• \(y = x' \sin \theta + y' \cos \theta\)

By applying these transformations, you align the coordinate system with features of the conic, like its axes. This is crucial for converting complex conic equations into a simpler form.

Polar Form

Polar coordinates provide a way to represent points in a plane using a distance and an angle, rather than the traditional cartesian coordinates. The polar form focuses on the center of the coordinate system, making it easier to express some mathematical equations, such as those involving circular and rotational symmetries.

The connection between rectangular coordinates \( (x, y) \) and polar coordinates \((r, \theta)\) can be established by:

• \(x = r\cos\theta\)
• \(y = r\sin\theta\)

This transformation allows you to express equations, like the given conic equation \(xy=2\), in terms of \(r\) (distance) as a function of \(\theta\) (angle). Transforming from rectangular to polar involves substituting these expressions and simplifying as shown in the solution.

Rectangular Coordinates

Rectangular coordinates use the familiar \(x\) and \(y\) axes for creating a grid-like system, where each point is determined by horizontal and vertical distances from the origin. This system is widely used and understood, making it intuitive for solving many typical mathematical problems.

In solving the problem of converting \(xy = 2\) into polar form, the original equation is expressed in rectangular coordinates. The challenge is converting it to a form based on distances and angles (polar coordinates). Remember:

• The \(x\) component is represented as \(r\cos\theta\)
• The \(y\) component as \(r\sin\theta\)

This transition lets us leverage the strengths of both coordinate systems to simplify solving or analyzing geometrical problems, especially those involving symmetry and circles.