Problem 24 - Thomas Calculus

Question

Find polar equations for the circles in Exercises \(21-28 .\) Sketch each circle in the coordinate plane and label it with both its Cartesian and polar equations. $$ x^{2}+(y+7)^{2}=49 $$

Step-by-Step Solution

Answer

Polar equation: \(r = -14\sin\theta\). Cartesian equation: \(x^2 + (y + 7)^2 = 49\). Center: \((0, -7)\), Radius: 7.

Step 1: Recognize the equation form

The given equation \(x^2 + (y + 7)^2 = 49\) is in the standard form of a circle \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. In this equation, \(h = 0\) and \(k = -7\), and \(r = 7\). So, the center is \((0, -7)\) and the radius is 7.

Step 2: Express circle in polar coordinates

In polar coordinates, a circle is given by \(r = 2a\cos(\theta)\) or \(r = 2a\sin(\theta)\) for horizontal and vertical alignments, respectively. This circle is centered at \((0, -7)\), so it is vertically aligned. Let's convert this Cartesian equation into a polar form: First, rewrite \(x^2 + (y + 7)^2 = 49\) to \(x^2 + y^2 + 14y + 49 = 49\). Reducing gives \(x^2 + y^2 + 14y = 0\). In polar form, \(x^2 + y^2 = r^2\) and \(y = r\sin\theta\), therefore: \[ r^2 + 14r \sin\theta = 0 \] \[ r(r + 14\sin\theta) = 0 \] The nontrivial solution is \(r = -14\sin\theta\).

Step 3: Finalize polar equation

The polar equation of the circle is \(r = -14\sin\theta\). This correctly represents a circle centered below the origin on the polar plane, which agrees with the center \((0, -7)\) found in Cartesian terms.

Step 4: Sketch the circle

Draw the coordinate axes. Plot the center of the circle at \((0, -7)\) in Cartesian coordinates. Sketch a circle with radius 7 around this center. On this sketch, note the polar equation \(r = -14\sin\theta\) and the Cartesian equation \(x^2 + (y + 7)^2 = 49\).

Key Concepts

circle equation

The equation \(x^2 + (y + 7)^2 = 49\) represents a circle in Cartesian coordinates.
When you see an equation in the form \((x-h)^2 + (y-k)^2 = r^2\), recognize it as the equation of a circle.
Here:

• \(h\) and \(k\) are the x and y coordinates of the center.
• \(r\) is the radius of the circle.

In our equation, substituting values gives:

• Center \((h, k) = (0, -7)\)
• Radius \(r = 7\)

Identifying the center and radius helps in both sketching the circle and converting it to another coordinate system.
It provides a simpler understanding of how the circle is positioned on the Cartesian plane, laying the groundwork for further analysis in different coordinate systems.

converting Cartesian to polar

To express the circle equation \(x^2 + (y+7)^2 = 49\) in polar coordinates, we need to transform these variables.
Transformation is based on:

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

Substituting into the given equation, we derive:\[r^2 + 14r\sin\theta = 0\]Factoring out \(r\), the result is:\[r(r + 14\sin\theta) = 0\]Here, the nontrivial solution is \(r = -14\sin\theta\).
This equation means that the circle's location and shape can now be comprehended in a polar context.
It's an effective way to understand circles from different perspectives, especially if points are calculated based on angular measurements.

polar sketching

When you're sketching a circle based on polar coordinates, use the polar equation \(r = -14\sin\theta\).
This helps visualize the circle on a polar diagram.
Essential steps:

• Draw the polar axes as two perpendicular lines.
• The center is effectively shifted along the negative y-axis, which aligns with both the given center \((0, -7)\) in Cartesian coordinates and this polar equation.
• The radius \(7\) simplifies to a consistent size in both coordinates.
• Plot points at varying \(\theta\) values if needed to understand the full sketch and dimensions of the circle, based on a radius of 7 units from the determined center.

Labeling the circle with both the polar \(r = -14\sin\theta\) and Cartesian \(x^2 + (y+7)^2 = 49\) equations helps reinforce the relationship between the two coordinate systems.
By combining these perspectives, you gain comprehensive insights into the shape and size of the circle.
This aids in applying the concept of polar coordinates to solve practical geometric problems.