Problem 29 Convert each conic into rectangu... [FREE SOLUTION]

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Precalculus Functions and Graphs

Mark Dugopolski

$Math Studyset 91影视 Explanations$ Math

4 Edition

Chapter 10: Problem 29

Convert each conic into rectangular coordinates and identify the conic. $$r=\frac{2}{6-\sin \theta}$$

Short Answer

Expert verified

The conic is an ellipse represented by the equation $36x^2 + 35y^2 - 4y - 4 = 0$.

Step by step solution

Identify Polar Equation

The given polar equation is: \[ r = \frac{2}{6 - \sin \theta} \]

Use Relationship Between Polar and Rectangular Coordinates

Recall the relationships: \[ r^2 = x^2 + y^2 \] \[ x = r \cos \theta \] \[ y = r \sin \theta \] Substitute these into the given equation.

Rewrite in Terms of r and $\theta$

Rewrite the equation as: \[ r(6 - \sin \theta) = 2 \] which simplifies to: \[ 6r - r\sin \theta = 2 \]

Substitute for $ r\sin \theta $

Considering that $ y = r \sin \theta $, the equation becomes: \[ 6r - y = 2 \]

Solve for r

Isolate r on one side: \[ 6r = y + 2 \] \[ r = \frac{y + 2}{6} \]

Substitute $ r = \sqrt{x^2 + y^2} $

Substitute $ r $ back into the equation: \[ \sqrt{x^2 + y^2} = \frac{y + 2}{6} \]

Square Both Sides

Square both sides to remove the square root: \[ x^2 + y^2 = \left( \frac{y + 2}{6} \right)^2 \] which expands to: \[ x^2 + y^2 = \frac{(y + 2)^2}{36} \]

Simplify the Equation

Multiply both sides by 36 to simplify: \[ 36(x^2 + y^2) = (y + 2)^2 \] Then expand and simplify: \[ 36x^2 + 36y^2 = y^2 + 4y + 4 \] Combine like terms: \[ 36x^2 + 35y^2 - 4y - 4 = 0 \]

Determine the Type of Conic

The general form of the equation indicates it's an ellipse because the coefficients of $x^2$ and $y^2$ are both positive but not equal.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polar Coordinates

Polar coordinates are a way of representing points in a plane using a distance and an angle. Instead of using the traditional Cartesian (rectangular) coordinates (x, y), we use a radius (\r\rm) and an angle (\theta):

$r$ is the distance from the origin to the point.
$\theta$ is the angle formed with the positive x-axis.

This system is particularly useful for problems with circular symmetry, such as those involving conic sections.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are the more familiar system using (x, y) to describe a point in a plane. Each point is located based on its horizontal (x) and vertical (y) distances from the origin. To convert between polar and rectangular coordinates, the key relationships are: \[ r^2 = x^2 + y^2 \] \[ x = r \cos \theta \] \[ y = r \sin \theta \] These formulas are the basis for converting an equation from polar to rectangular form.

Conic Sections

Conic sections are the curves obtained by intersecting a cone with a plane. The main types include:

Ellipses
Parabolas
Hyperbolas
Circles (a special type of ellipse)

Each type of conic section can be represented by a specific polar or rectangular equation. Recognizing the form of these equations helps to identify the conic.

Ellipse Equations

An ellipse equation in rectangular coordinates typically looks like this: \[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \] However, elliptical equations sometimes take other forms, such as the one we derived in the example problem: \[ 36x^2 + 35y^2 - 4y - 4 = 0 \] By identifying key characteristics in the converted rectangular form (like the coefficients), we confirmed it as an ellipse due to positive but unequal coefficients of $x^2$ and $y^2$.

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91影视

Convert each conic into rectangular coordinates and identify the conic. $$r=\frac{2}{6-\sin \theta}$$

Short Answer

Step by step solution

Identify Polar Equation

Use Relationship Between Polar and Rectangular Coordinates

Rewrite in Terms of r and \(\theta\)

Substitute for \( r\sin \theta \)

Solve for r

Substitute \( r = \sqrt{x^2 + y^2} \)

Square Both Sides

Simplify the Equation

Determine the Type of Conic

Key Concepts

Polar Coordinates

Rectangular Coordinates

Conic Sections

Ellipse Equations

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