Problem 20 - Essential Calculus

Question

Convert the rectangular equation to polar form and sketch its graph. $$ x^{2}+y^{2}-2 a x=0 $$

Step-by-Step Solution

Answer

The polar form of the given rectangular equation is \(r(r-2a \cdot cos(θ)) = 0\), and its graph includes a single point at the origin and a circle centered at (a,0) with a radius a.

Step 1: Substitute in the values of x and y

Use the formulas \(x = r \cdot cos(θ)\), \(y = r \cdot sin(θ)\) and substitute in the provided equation. The rectangular equation \(x^{2} + y^{2} - 2ax = 0\) turns into \((r \cdot cos(θ))^{2} + (r \cdot sin(θ))^{2} - 2a \cdot r \cdot cos(θ) = 0\).

Step 2: Simplify the equation

Now, simplify this equation by applying the identities. The identity \(sin^2(θ) + cos^2(θ) = 1\) is helpful in this context. The above equation simplifies to \(r^{2}-2a \cdot r \cdot cos(θ) = 0\).

Step 3: Factoring the simplified equation

Factor out \(r\) from the equation to arrive at the final polar equation. The factored equation will become \(r(r-2a \cdot cos(θ)) = 0\).

Step 4: Sketching the graph

This is the equation of a circle in polar coordinates. The impact made by the \(r = 0\) term is just a single point at the origin. The other factor \(r-2a \cdot cos(θ) = 0\) can be rewritten as \(r = 2a \cdot cos(θ)\) which is a circle centered at (a,0) with radius a. The graph of this equation will consist of this circle together with the point at the origin.