Question A.3: Identifying and Converting from Polar Form to Rectangular Fo...
Identifying and Converting from Polar Form to Rectangular Form
Identify the type of conic represented by r = \frac{8}{2 – \cos θ} . Then convert the equation to rectangular form.
To identify the type of conic, we divide both the numerator and the denominator on the right side of the equation by 2.
r = \frac{4}{1 – \frac{1}{ 2} \cos θ}
From the table, we see that this is a conic that has a vertical directrix, with e = \frac{1}{ 2} , making it an ellipse. To convert to rectangular form, we start with the given equation.
r = \frac{8}{2 – \cos θ} Given equation
r (2 – \cosθ)= 8 Multiply by 2 – \cos θ.
2r – r \cosθ = 8 Distributive property
2r = r \cosθ + 8 Add r \cosθ to each side.
(2r)² = (r\cosθ + 8)² Square each side.
(2r)²= (x + 8)² r \cosθ = x
4r² = x² + 16x + 64 Multiply.
4(x²+ y²) = x²+ 16x + 64 r² = x² + y²
4x² + 4y² = x²+ 16x + 64 Distributive property
3x² + 4y² – 16x – 64 = 0 Standard form
The coefficients of x² and y² are both positive and are not equal, further supporting our assertion that the graph is an ellipse.