Question

transform the polar equation to an equation in rectangular coordinates. then identify and graph the equation. ( r = - 14 cos \theta ) write an equation in rectangular coordinates. ( ( x + 7 ) ^ { 2 } + y ^ { 2 } = 49 ) (type an equation ) what is the graph of this equation? a. vertical line b. horizontal line c. circle with center at ( ( - 7,0 ) ) d. circle with center at ( ( 7,0 ) )

Explanation:

Step1: Recall the conversion formulas

We know that \(x = r\cos\theta\) and \(r^{2}=x^{2}+y^{2}\). Given \(r=- 14\cos\theta\), multiply both sides by \(r\): \(r^{2}=-14r\cos\theta\).

Step2: Substitute the conversion formulas

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

Step3: Complete the square

Rearrange the equation \(x^{2}+y^{2}=-14x\) to \(x^{2}+14x + y^{2}=0\). Completing the square for the \(x\) - terms: \((x + 7)^{2}-49+y^{2}=0\), which simplifies to \((x + 7)^{2}+y^{2}=49\).

Step4: Identify the graph

The general equation of a circle is \((x - a)^{2}+(y - b)^{2}=R^{2}\), where \((a,b)\) is the center and \(R\) is the radius. For the equation \((x + 7)^{2}+y^{2}=49=(7)^{2}\), the center is \((-7,0)\) and the radius \(R = 7\).

Answer:

C. circle with center at \((-7,0)\)