Problem 38 - Algebra and Trigonometry Real Mathematics, Real People

Question

Eliminate the parameter and obtain the standard form of the rectangular equation. Circle: \(x=h+r \cos \theta, y=k+r \sin \theta\)

Step-by-Step Solution

Answer

The standard form of the rectangular equation of the circle is \((x-h)^2 + (y-k)^2 = r^2\).

Step 1: Express \(x-h\) and \(y-k\) in terms of \(r\)

Start by expressing \(x-h\) and \(y-k\) as: \(x-h = r \cos \theta\) and \(y-k = r \sin \theta\), isolating the cosine and sine terms.

Step 2: Square each equation

Square both equations to obtain: \((x-h)^2 = (r \cos \theta)^2\) and \((y-k)^2 = (r \sin \theta)^2\). This gives: \((x-h)^2 = r^2 \cos^2 \theta\) and \((y-k)^2 = r^2 \sin^2 \theta\)

Step 3: Add the squared equations

Add the two equations to express the left-hand side as a sum of squares: \((x-h)^2 + (y-k)^2 = r^2 \cos^2 \theta + r^2 \sin^2 \theta\)

Step 4: Apply the Pythagorean identity

Use the Pythagorean identity \(\cos^2{\theta} + \sin^2{\theta}=1\) to replace the right-hand side of the equation, resulting in: \((x-h)^2 + (y-k)^2 = r^2( \cos^2 \theta + \sin^2 \theta) = r^2\)