Problem 22 - College Algebra

Question

Convert the polar equation of a conic section to a rectangular equation. $$ r=\frac{4}{2+2 \sin \theta} $$

Step-by-Step Solution

Answer

The rectangular equation is \( x = 0 \).

Step 1: Convert to Standard Polar Form

The given polar equation is \( r = \frac{4}{2 + 2\sin\theta} \). Let's first rewrite it in the standard form \( r = \frac{ed}{1 - e\sin\theta} \) for a conic section. Factoring out a 2 in the denominator gives: \[ r = \frac{4}{2(1 + \sin\theta)} = \frac{2}{1 + \sin\theta}. \] This indicates that \( e = -1 \) and \( ed = 2 \).

Step 2: Identify Eccentricity and Directrix

We have \( e \) identified as \( -1 \), meaning it's an eccentricity corresponding to a parabola, not typical for this case due to sine. Now looking at typical eccentricities (0 < e < 1 for ellipse, e=1 for parabola, e > 1 for hyperbola), corrected it's ellipse (as negative reversed here hint errors unless swapped sign sense for use sake). Given the needs direct formula sense, correct to use \( e=1 \).

Step 3: Polar to Rectangular Conversion

We'll convert the polar form to rectangular coordinates using the identities: \( x = r\cos\theta \) and \( y = r\sin\theta \), along with \( r^2 = x^2 + y^2 \). Multiply the equation by \( r \): \[ 2r = r + r^2\sin\theta. \] Substitute \( r = \sqrt{x^2 + y^2} \) and \( r\sin\theta = y \) to obtain: \[ 2\sqrt{x^2 + y^2} = \sqrt{x^2 + y^2} + y. \]

Step 4: Simplification

Isolate the terms involving \( sqrt{x^2+y^2} \): \[ \sqrt{x^2 + y^2} = y. \] Square both sides to remove the square root: \[ x^2 + y^2 = y^2. \] Simplify further by subtracting \( y^2 \) from both sides: \[ x^2 = 0. \] This implies \( x = 0 \).

Step 5: Conclusion

The final rectangular equation is \( x = 0 \). This indicates that the conic section is a vertical line along the y-axis or more so since converted here it means transforms as must comprehend using reverted-form beyond just direct conversion expectations here.

Key Concepts

Conic Sections

Conic sections are shapes created by the intersection of a plane and a cone. They include ellipses, parabolas, circles, and hyperbolas. The type of curve produced depends on how the plane slices through the cone.

• An ellipse results when the plane cuts through the cone at an angle, but doesn't intersect the base.
• A parabola occurs when the plane is parallel to one of the cone's sides.
• A hyperbola forms when the plane cuts through both halves of the double cone.
• A circle is a special ellipse with equal axes created by slicing the cone parallel to its base.

Understanding conic sections helps in graphing equations that describe other natural phenomena, like planetary orbits which are ellipses.

Eccentricity

Eccentricity is a measure of how "stretched" a conic section is. It's a number that describes the shape of the curve:

• A circle has an eccentricity of 0, indicating no stretch.
• An ellipse has an eccentricity between 0 and 1.
• A parabola has an eccentricity of exactly 1, showing a neat balance of curve.
• A hyperbola has an eccentricity greater than 1, indicating a wide opening.

Eccentricity helps in identifying the specific type of conic section and its geometry.

Rectangular Equation Conversion

Converting a polar equation to a rectangular equation involves using trigonometric identities and algebraic manipulation. Here's the general process for conversion:1. Express polar coordinates in terms of rectangular coordinates: - Use the formulas: \( x = r\cos\theta \) \( y = r\sin\theta \)2. Substitute these expressions into the polar equation.3. Simplify the resulting equation into standard rectangular form (typically involving only x and y).In the given exercise, applying these steps led us to the equation \( x = 0 \), which represents a line along the y-axis in rectangular coordinates.

Polar Equations

Polar equations express relationships using polar coordinates \((r, \theta)\), where \(r\) is the radius (distance from the origin) and \(\theta\) is the angle from the positive x-axis.

• They are especially useful in modeling circular and rotational objects.
• Transforming between polar and rectangular coordinates depends on the trigonometric functions of sine and cosine.
• Common conversions involve using equations like: - \( x = r\cos\theta \) - \( y = r\sin\theta \) - \( r^2 = x^2 + y^2 \)

In the exercise, we converted a polar equation for a conic into a rectangular equation, uncovering its geometric significance in a different coordinate system.