Question

1. sketch the graph of the following polar equation.

$r = -14cos\theta$

Explanation:

Step1: Recall Polar to Cartesian Conversion

We know the relationships \( x = r\cos\theta \), \( y = r\sin\theta \), and \( r^{2}=x^{2}+y^{2} \). Start with the given polar equation \( r=- 14\cos\theta \). Multiply both sides by \( r \) to get \( r^{2}=-14r\cos\theta \).

Step2: Substitute Cartesian Equivalents

Substitute \( r^{2}=x^{2}+y^{2} \) and \( r\cos\theta = x \) into the equation. We obtain \( x^{2}+y^{2}=-14x \).

Step3: Complete the Square

Rearrange the equation: \( x^{2}+14x + y^{2}=0 \). To complete the square for the \( x \)-terms, take half of 14 (which is 7), square it (49), and add it to both sides: \( x^{2}+14x + 49+y^{2}=49 \). This factors to \( (x + 7)^{2}+y^{2}=49 \).

Step4: Identify the Graph

The equation \( (x + 7)^{2}+y^{2}=49 \) is in the standard form of a circle \((x - h)^{2}+(y - k)^{2}=r^{2}\), where the center is \((h,k)=(-7,0)\) and the radius \( r = 7 \). To sketch it, plot the center at \((-7,0)\) on the Cartesian plane and draw a circle with radius 7. In polar coordinates, since the equation is of the form \( r = a\cos\theta \) (with \( a=- 14 \), but the general form for a circle centered on the \( x \)-axis) , we can also analyze key points. When \( \theta = 0 \), \( r=-14\cos(0)=-14 \) (but in polar coordinates, \( r=-14 \) at \( \theta = 0 \) is equivalent to \( r = 14 \) at \( \theta=\pi \), so the point is \((-14,0)\) or \((14,\pi)\)). When \( \theta=\frac{\pi}{2} \), \( r=-14\cos(\frac{\pi}{2}) = 0 \), so the point is at the origin. When \( \theta=\pi \), \( r=-14\cos(\pi)=14 \), so the point is \((14,\pi)\) or \((-14,0)\) (consistent with before). When \( \theta=\frac{3\pi}{2} \), \( r=-14\cos(\frac{3\pi}{2}) = 0 \), again at the origin. Connecting these key points and using the circle equation, we can sketch the graph.

Answer:

The graph of \( r=-14\cos\theta \) is a circle with center \((-7,0)\) (or in polar terms, centered at \( r = 7 \), \( \theta=\pi \)) and radius 7. The sketch is a circle centered at \( x=-7,y = 0 \) with radius 7.