Polar Coordinates
The line \(y = ax + b\) in Cartesian coordinates can be written as \[r = \frac{13}{\sin \theta - 24 \cos \theta}\] in polar coordinates. Assuming \(b\) is positive, what is the value of \(a + b\)?
The graph \(r= \frac{2}{1-6\sin \theta}\) in polar coordinates can be expressed as \(x^2 = ay^2+by+c\) in Cartesian coordinates, where \(a\), \(b\) and \(c\) are real numbers. What is the value of \(a+b+c\)?
Letting \(\theta\) vary from \(0\) to \(2\pi,\) the parametric equations \(x= 5 + \cos \theta\) and \(y = \sin \theta\) represent a circle \(\Gamma_1.\) There is a circle \(\Gamma_2\) that is externally tangent to \(\Gamma_1,\) tangent to the \(y\)-axis, and centered (in Cartesian coordinates) at \((10, \sqrt{a}).\) What is the value of \(a?\)
If \(x\) and \(y\) satisfy \(x=3 \cos \theta\) and \(y = 8 + 3 \sin \theta\), the graph of \((x, y)\) is a circle. If the center of this circle is \((a, b)\) and the radius is \(r\), what is the value of \(a+b+r\)?
Which of the following Cartesian coordinate equations represents the polar equation \(r=14 \cos \theta\ (0 \leq \theta < 2\pi)?\)