Problem 19 - College Algebra

Question

Convert the polar equation of a conic section to a rectangular equation. $$ r=\frac{2}{5-3 \sin \theta} $$

Step-by-Step Solution

Answer

The rectangular equation is \( 5(x^2 + y^2) - 3\sqrt{x^2 + y^2}y = 2 \).

Step 1: Identify the Equation of the Conic

The given equation is in the form \( r = \frac{ed}{1 - e\sin(\theta)} \), which is the polar form of a conic section. It appears that this form has been altered to match the equation \( r = \frac{2}{5 - 3 \sin(\theta)} \), where \( e \) is the eccentricity, and \( d \) is the semi-latus rectum. Here, \( e \) and \( d \) need to be identified: \( e = 3/5 \) and \( ed = 2 \).

Step 2: Solve for Semi-Latus Rectum \(d\)

Using the equation \( ed = 2 \), and knowing \( e = \frac{3}{5} \), solve for \( d \): \[ \left(\frac{3}{5}\right)d = 2 \] Multiply both sides by \( \frac{5}{3} \): \[ d = \frac{10}{3} \].

Step 3: Express \( r \) in Terms of \( x \) and \( y \)

Recall the polar-to-rectangular transformations: \( r = \sqrt{x^2 + y^2} \) and \( \sin \theta = \frac{y}{r} \). Substitute \( \sin \theta = \frac{y}{r} \) into the polar equation: \[ r = \frac{2}{5 - 3\left(\frac{y}{r}\right)} \] Multiply both sides by \( 5r - 3y \): \[ r(5r - 3y) = 2 \].

Step 4: Expand and Simplify

Distribute \( r \) to get: \[ 5r^2 - 3ry = 2 \]. Express \( r^2 \) using \( x \) and \( y \) as \( r^2 = x^2 + y^2 \), and substitute this into the equation: \[ 5(x^2 + y^2) - 3ry = 2 \].

Step 5: Eliminate \( r \) from the Equation

We know \( r = \sqrt{x^2 + y^2} \), so substitute it in: \[ 5(x^2 + y^2) - 3\sqrt{x^2 + y^2}y = 2 \]. This is the rectangular form of the given polar equation.

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. Depending on the angle of the plane relative to the cone, conic sections can take the form of a circle, ellipse, parabola, or hyperbola. Each of these shapes has distinct properties:

• Circle: A special case of an ellipse where the eccentricity is 0. All points on the circle are equidistant from the center.
• Ellipse: An elongated circle with eccentricity between 0 and 1. The sum of the distances from the foci to any point on the ellipse is constant.
• Parabola: A curve where each point is equidistant from a fixed point (focus) and a line (directrix). It has an eccentricity of 1.
• Hyperbola: A shape with two separate branches, which occurs when the plane intersects both nappes of the cone. It has an eccentricity greater than 1.

Conic sections are described by quadratic equations in rectangular coordinates, but they can also be represented in polar form. This dual representation makes them very versatile for different types of mathematical modeling.

Polar Coordinates

Polar coordinates provide a different way of describing the location of a point compared to the usual rectangular (or Cartesian) coordinates. Instead of using x and y coordinates, polar coordinates use:

• r (radius): The distance from the origin to the point.
• \( \theta \) (angle): The angle from the positive x-axis to the radius line connecting the origin to the point.

The transformation from polar to rectangular coordinates relies on simple relationships:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)
These formulas allow you to convert coordinates from polar to rectangular systems, which is particularly useful in scenarios where rotational symmetry is present, such as in the study of conic sections.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are the most common coordinate system used in math. They are simply points on a flat plane defined by two numbers: x and y. These coordinates describe horizontal and vertical distances from the origin (0,0). Each point in the plane is represented by an ordered pair:

• x-coordinate: How far left or right the point is from the origin.
• y-coordinate: How far up or down the point is from the origin.

In problems involving conversions, rectangular coordinates are crucial for translating equations from polar form. The process involves replacing polar parameters (r and \( \theta \)) with their rectangular equivalents \( \sqrt{x^2 + y^2} \) and \( \sin \theta = \frac{y}{r} \). This is the basis of the conversion seen in the problem, where a polar equation is transformed into a rectangular one, allowing for easier analysis and graphical representation of the conic sections.