Problem 37 - Thomas Calculus

Question

Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian equations. Then describe or identify the graph. $$r=\frac{5}{\sin \theta-2 \cos \theta}$$

Step-by-Step Solution

Answer

Convert the equation to Cartesian by substituting polar relationships, resulting in \((x^2 + y^2)(y - 2x)^2 = 25\), identifying it as a conic.

Step 1: Understand Polar to Cartesian Conversion

Recall the relationships between polar coordinates \((r, \theta)\) and Cartesian coordinates \((x, y)\):\[ x = r\cos\theta, \quad y = r\sin\theta, \quad r^2 = x^2 + y^2. \] To convert a polar equation to Cartesian form, we will need to replace \(r\), \(\sin \theta\), and \(\cos \theta\) with their Cartesian equivalents using these relationships.

Step 2: Substitute Trigonometric Functions

We know that \(\sin\theta = \frac{y}{r}\) and \(\cos\theta = \frac{x}{r}\). Substitute these into the equation:\[ r = \frac{5}{\frac{y}{r} - 2 \cdot \frac{x}{r}}. \]This becomes:\[ r = \frac{5}{\frac{y - 2x}{r}}. \]

Step 3: Simplify the Equation

Multiply both sides by \(y - 2x\) to eliminate the fraction:\[ r \cdot (y - 2x) = 5. \]Substitute \(r = \sqrt{x^2 + y^2}\) into the equation:\[ \sqrt{x^2 + y^2} (y - 2x) = 5. \]

Step 4: Square Both Sides

To eliminate the square root, square both sides of the equation:\[ (x^2 + y^2)(y - 2x)^2 = 5^2. \]This is a key step in rewriting the equation completely in terms of \(x\) and \(y\).

Step 5: Simplify Further

Expand \((y - 2x)^2\) as \((y - 2x)(y - 2x) = y^2 - 4xy + 4x^2\), and rewrite the equation:\[ (x^2 + y^2)(y^2 - 4xy + 4x^2) = 25 \]

Step 6: Describe and Identify the Graph

The equation involves the terms \((y^2 - 4xy + 4x^2)\) multiplied by \((x^2 + y^2)\), suggesting that it's an equation of a conic section. By analyzing the squared terms and the general form, this is an equation of a circle or another type of conic without cross-product terms and offsets, making rigorous analysis needed to define the graph precisely.

Key Concepts

Polar Coordinates

Polar coordinates are a powerful way to describe the location of a point in the plane.They use the distance from a reference point, and the angle from a reference direction. In polar coordinates, a point is represented as

• r: the radial distance from the origin (or pole).
• \(\theta\): the angle measured in radians from the positive x-axis.

This system is particularly useful for solving problems with circular or rotational symmetry as it simplifies the geometry.
To convert polar coordinates to Cartesian coordinates, we use the following relationships:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)
• \(r^2 = x^2 + y^2\)

By translating these geometric interpretations into algebraic expressions, one can move fluidly between polar and Cartesian representations depending on the needs of the problem.

Cartesian Coordinates

Cartesian coordinates use a system based on two perpendicular axes intersecting at a common origin. These axes divide the plane into four quadrants. In this coordinate system, the position of a point is given by

• x: the horizontal distance from the origin, viewed along the x-axis.
• y: the vertical distance from the origin, viewed along the y-axis.

This provides a straightforward way to graph lines, curves, and other geometric figures.
Some benefits of using Cartesian coordinates are that they allow easy computation of distances, slopes, and the formulation of geometric shapes. For instance, converting polar equations to Cartesian can help uncover familiar geometric forms like circles and lines.
In the previously provided solution, the transformation enabled the depiction of the polar equation as a Cartesian equation, making it easier to graph and interpret any corresponding geometric shape.

Conic Sections

Conic sections refer to the shapes created by intersecting a plane with a double-napped cone.These sections include circles, ellipses, parabolas, and hyperbolas.Each shape has unique properties and equations.
This concept is essential in mathematics due to its applications in fields like astronomy, physics, and engineering.

• Circle: Formed when the plane cuts the cone parallel to its base. Its Cartesian equation typically looks like \((x - h)^2 + (y - k)^2 = r^2\).
• Ellipse: Formed when the plane cuts all the generators of the cone without being parallel to the base. Its equation resembles \(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\).
• Parabola: Results when the plane is parallel to a generator of the cone. Its basic form is \(y = ax^2 + bx + c\).
• Hyperbola: Occurs when the plane intersects both halves of the double cone. Its form is \(\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1\).

The equation in the original exercise transforms into a complex algebraic form which suggests it might fit one of these categorizations.Identifying the specific conic type requires exploring the transformation results with care.