11. sketch the graph of the following polar equation.\n$$r = -14 \\cos \\theta$$
11. sketch the graph of the following polar equation.\n$$r = -14 \\cos \\theta$$
Answer
Explanation:
Step1: Identify the polar equation type
The equation $r = a \cos \theta$ represents a circle passing through the origin.
Step2: Convert to rectangular coordinates
Multiply both sides by $r$ and use $r^2 = x^2 + y^2$ and $x = r \cos \theta$. $$x^2 + y^2 = -14x$$
Step3: Complete the square for $x$
Rearrange the equation and add $(\frac{14}{2})^2$ to both sides. $$x^2 + 14x + 49 + y^2 = 49$$
Step4: Determine circle properties
Write in standard form $(x - h)^2 + (y - k)^2 = R^2$. $$(x + 7)^2 + y^2 = 7^2$$
Step5: Describe the graph
The graph is a circle with center $(-7, 0)$ and radius $7$.
Answer:
The graph is a circle centered at $(-7, 0)$ in the rectangular coordinate system (or $r=7, \theta=\pi$ in polar) with a radius of $7$. It lies entirely on the left side of the vertical axis (pole), passing through the origin and the point $(-14, 0)$.