Problem 43 - 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. $$r^{2}+2 r^{2} \cos \theta \sin \theta=1$$

Step-by-Step Solution

Answer

The Cartesian equation is \(x^2 + y^2 + 2xy = 1\), representing an ellipse.

Step 1: Recall Polar to Cartesian Conversion

To convert from polar coordinates \(r, \theta\) to Cartesian coordinates \(x, y\), we use the relationships \(x = r \cos \theta\) and \(y = r \sin \theta\). Also, remember \(r^2 = x^2 + y^2\).

Step 2: Identify Terms in the Given Equation

The given polar equation is \(r^2 + 2r^2 \cos \theta \sin \theta = 1\). We need to convert each term into Cartesian coordinates using our conversions.

Step 3: Convert \\(r^2\\) to Cartesian Form

Using the identity \(r^2 = x^2 + y^2\), replace \(r^2\) with \(x^2 + y^2\) in the equation. So, it becomes \(x^2 + y^2 + 2r^2 \cos \theta \sin \theta = 1\).

Step 4: Convert \\(r^2 \cos \theta \sin \theta\\) to Cartesian Form

Recall \(\cos \theta = \frac{x}{r}\) and \(\sin \theta = \frac{y}{r}\). Thus, \(r^2 \cos \theta \sin \theta = r^2 \left(\frac{x}{r}\right)\left(\frac{y}{r}\right) = xy\).

Step 5: Substitute Cartesian Terms into the Equation

Substitute \(x^2 + y^2\) for \(r^2\) and \(xy\) for \(2r^2 \cos \theta \sin \theta\) into the equation. The equation becomes \(x^2 + y^2 + 2xy = 1\).

Step 6: Identify the Cartesian Equation

The final equivalent Cartesian equation is \(x^2 + y^2 + 2xy = 1\). This equation represents a rotated conic section. By completing further analysis, it can be shown to be an ellipse after appropriate rotation transformation.

Key Concepts

Polar Coordinates

Polar coordinates are an excellent way to describe points in a plane, especially when dealing with circular or rotational systems. Instead of using the tradtional \(x, y\) coordinate system, polar coordinates make use of a radial distance, \(r\), and an angle, \(\theta\), which is measured from the positive x-axis. This makes it easier to describe curves and shapes that radiate around a point or have some symmetry related to a circle or angle.

A key feature of polar coordinates is the direct link between distance and direction:

• \(r\) represents how far away the point is from the origin.
• \(\theta\) (theta) indicates the angle direction of the point from the positive x-axis.

These parameters can be used to express various curves by varying the relationship between \(r\) and \(\theta\). One of the advantages of polar coordinates is their use in converting to Cartesian coordinates for analysis of equations and shapes.

Cartesian Coordinates

Cartesian coordinates are based on a straightforward, right-angle grid system, where any point on the plane is depicted by an ordered pair \(x, y\). Here, \(x\) is the horizontal distance from the origin, and \(y\) is the vertical distance.

This rectangular coordinate system is extremely effective for equations involving vertical and horizontal lines, parabolas, and other polynomial graphs. Its key elements include:

• Alignment along perpendicular axes.
• Simplicity in computing distances and slopes.

Transforming polar coordinates to Cartesian allows us to analyze the same point or equation in a typical graphing setting. This conversion aids in transforming complex polar equations into more familiar algebraic forms using the identities:

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

Such transformations make it easier to visualize and solve in everyday coordinate geometry.

Conic Sections

Conic sections are curves obtained by intersecting a cone with a plane at different angles. These geometric figures include circles, ellipses, parabolas, and hyperbolas. Each form relates to how a particular type of cutting of a cone can produce a different shape.

These sections play an essential role in various mathematical and scientific applications, such as astronomy, engineering, and physics. A few key characteristics include:

• Ellipses are formed by a plane cutting through both nappes of the cone.
• Parabolas result from the plane being parallel to the cone's lateral surface.
• Hyperbolas occur when such a plane cuts through both halves, forming two separate curves.

In Cartesian terms, conic sections can be represented by quadratic equations. The challenge of analyzing conics often includes identifying the nature of the conic and its position on a plane. For example, if part of a polar equation's conversion leads to an equation like \(x^2 + y^2 + 2xy = 1\), it can signify a transformed appearance of a conic section like an ellipse, which allows further geometrical analysis.