Question

1. sketch the graph of the following polar equation.

$r = -14\cos\theta$

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. So we have \( x^{2}+y^{2}=-14x \).

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. So \( x^{2}+14x + 49+y^{2}=49 \). This factors to \( (x + 7)^{2}+y^{2}=49 \).

Step4: Identify the Conic Section

The equation \( (x + 7)^{2}+y^{2}=49 \) is the standard form of a circle with center \( (-7,0) \) (in Cartesian coordinates) and radius \( 7 \). In polar coordinates, the general form of a circle with diameter along the polar axis (x - axis in Cartesian) and center at \( (r_0,\theta_0) \) can be related. For \( r = a\cos\theta \) (or \( r=-a\cos\theta \)), it's a circle. Here, since we have \( r=-14\cos\theta \), converting to Cartesian shows it's a circle with center at \( (-7,0) \) and radius 7. To sketch, plot the center at \( (-7,0) \) (in polar, this is a distance of 7 from the origin in the negative x - direction) and draw a circle with radius 7.

Answer:

The graph of \( r=-14\cos\theta \) is a circle with center at \( (-7,0) \) (in Cartesian coordinates) and radius 7. The sketch would show a circle centered 7 units to the left of the origin (in the polar coordinate system, this corresponds to a circle along the polar axis with the center on the negative x - axis relative to the origin) with radius 7.