Question

1. sketch the graph of the following polar equation.

\\(cos \theta = \frac{6}{r}\\)

Explanation:

Step1: Rewrite the polar equation

Start with the given polar equation \(\cos\theta=\frac{6}{r}\). Multiply both sides by \(r\) to get \(r\cos\theta = 6\).

Step2: Convert to rectangular coordinates

Recall the conversion formula \(x = r\cos\theta\) in polar - rectangular coordinate conversion. Substituting \(x=r\cos\theta\) into the equation \(r\cos\theta = 6\), we obtain the rectangular equation \(x = 6\).

Step3: Analyze the graph of the rectangular equation

The equation \(x = 6\) represents a vertical line in the rectangular (Cartesian) coordinate system. In the polar coordinate system, for a vertical line \(x = a\) (\(a>0\)), the polar equation (after conversion) \(r\cos\theta=a\) (or in the form we started with \(\cos\theta=\frac{a}{r}\)) has a graph that is a vertical line parallel to the \(y\) - axis, passing through the point \((6,0)\) in the rectangular coordinate system. To sketch it, we can note that for any value of \(\theta\) (except when \(\cos\theta = 0\), but in our case, from \(r\cos\theta=6\), \(r=\frac{6}{\cos\theta}\), and \(\cos\theta
eq0\) because if \(\cos\theta = 0\), the right - hand side \(\frac{6}{r}\) would not be zero), the \(x\) - coordinate of the point \((r\cos\theta,r\sin\theta)\) is always 6. So we draw a vertical line that intersects the \(x\) - axis at \((6,0)\) and is parallel to the \(y\) - axis.

Answer:

The graph of the polar equation \(\cos\theta=\frac{6}{r}\) (or \(r\cos\theta = 6\)) is a vertical line \(x = 6\) in the rectangular coordinate system (a vertical line parallel to the \(y\) - axis passing through \((6,0)\) in the \(xy\) - plane).