Question

the polar equation of the graph is either ( r = a + b cos \theta ) or ( a + b sin \theta ), ( a > 0 ), ( b > 0 ). select the correct equation and find the values of ( a ) and ( b ).
choose the correct form of the equation below.
( r = a + b cos \theta )
( r = a + b sin \theta )
what are the values of ( a ) and ( b )?
( a = )
( b = )

Explanation:

Step1: Analyze the symmetry

The graph is symmetric about the \(x -\)axis. For the polar equations \(r = a + b\cos\theta\) and \(r=a + b\sin\theta\), the equation \(r=a + b\cos\theta\) is symmetric about the \(x -\)axis (since \(\cos(-\theta)=\cos\theta\) and \(r(-\theta)=a + b\cos(-\theta)=a + b\cos\theta=r(\theta)\)), and the equation \(r=a + b\sin\theta\) is symmetric about the \(y -\)axis (since \(\sin(-\theta)=-\sin\theta\) and \(r(-\theta)=a + b\sin(-\theta)=a - b\sin\theta
eq r(\theta)\) in general). So the correct form is \(r=a + b\cos\theta\).

Step2: Substitute \(\theta = 0\) and \(\theta=\frac{\pi}{2}\)

When \(\theta = 0\), \(r=a + b\cos(0)=a + b\). Given the point \((9,0)\) (in polar coordinates \(r = 9\) when \(\theta = 0\)), so \(a + b=9\).
When \(\theta=\frac{\pi}{2}\), \(r=a + b\cos(\frac{\pi}{2})\). Since \(\cos(\frac{\pi}{2}) = 0\), and the point \((8,\frac{\pi}{2})\) (in polar coordinates \(r = 8\) when \(\theta=\frac{\pi}{2}\)), so \(a=8\).

Step3: Solve for \(b\)

Substitute \(a = 8\) into \(a + b=9\). We get \(8 + b=9\), then \(b=9 - 8=1\).

Answer:

\(a = 8\), \(b = 1\)