Problem 43 - Essential Calculus

Question

Convert the equation \(r=2(h \cos \theta+k \sin \theta)\) to rectangular form and verify that it is the equation of a circle. Find the radius and the rectangular coordinates of the center of the circle.

Step-by-Step Solution

Answer

The circle's center is at the point (h, k) and the radius of the circle is \(\sqrt{h^2 + k^2}\).

Step 1: Conversion from Polar to Rectangular

The equation \(r=2(h \cos \theta+k \sin \theta)\) is given in polar form. To convert this into rectangular coordinates, we can substitute \(r = \sqrt{x^2 + y^2}\), \(\cos \theta = x / r\), and \(\sin \theta = y / r\) into the given equation. This gives us \(\sqrt{x^2 + y^2} = 2(h (x / \sqrt{x^2 + y^2}) + k (y / \sqrt{x^2 + y^2}))\). This simplifies to \(\sqrt{x^2 + y^2} = 2hx/r + 2ky/r\). Multiply by \(\sqrt{x^2 + y^2}\) and simplify to get \(x^2 + y^2 = 2hx + 2ky\).

Step 2: Confirm equation as Circle

The general equation for a circle is \((x-a)^2 + (y-b)^2 = r^2\). To rewrite the equation \(x^2 + y^2 = 2hx + 2ky\) in that form, move the terms around to get \((x-h)^2 + (y-k)^2 = h^2 + k^2\). This is equivalent to the general equation of circle which confirms our circle.

Step 3: Find the Center and Radius

Comparing \((x-h)^2 + (y-k)^2 = h^2 + k^2\) to the general circle equation \((x-a)^2 + (y-b)^2 = r^2\), we can identify \(a = h\), \(b = k\), and \(r = \sqrt{h^2 + k^2}\) which gives us the center and radius of the circle.

Key Concepts

Circle Equation

Understanding the equation of a circle is fundamental in coordinate geometry. A circle's equation in standard form is \((x-a)^2 + (y-b)^2 = r^2\), where \((a, b)\) is the center and \(r\) is the radius.
This structure comes from the Pythagorean theorem, maintaining a constant distance from the center point to any point on the circle.
To rewrite an equation like \(x^2 + y^2 = 2hx + 2ky\) in this form, you rearrange and complete the square, resulting in \((x-h)^2 + (y-k)^2 = h^2 + k^2\).
This confirms it represents a circle, where the center is \((h, k)\) and the radius is \(\sqrt{h^2 + k^2}\). Recognizing the structure is key to identifying and working with circles in different settings.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, connects algebra and geometry through graphs and equations.
It's essential for translating shapes and curves into mathematical expressions.
When working with circles, lines, and other geometric figures, coordinate geometry gives us the tools to calculate intersections, distances, and areas by converting these figures into equations we can manipulate.

• Equations: Different equations represent different geometric shapes like lines, circles, and parabolas.
• Graphical Representation: Understanding the graphical implications helps to visualize solutions.

Mastering this is crucial for solving complex problems involving multiple geometric shapes.

Polar Coordinates

Polar coordinates provide a different way of describing a point's location. Instead of using \((x, y)\) as in the Cartesian system, it uses \((r, \theta)\), where \(r\) is the distance from the origin, and \(\theta\) is the angle from the positive x-axis.
This is particularly useful for circular and rotational problems.
Converting between polar and rectangular coordinates involves the relations:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)
• \(r = \sqrt{x^2 + y^2}\)
• \(\theta = \tan^{-1}(y/x)\)

This flexibility in representation helps solve problems that are more naturally aligned with circular symmetry, as seen in the problem of converting \(r = 2(h \cos \theta + k \sin \theta)\) into its rectangular form.