Algebra

# Polar Coordinates - Convert Functions

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 $$a+b+c$$?

In polar coordinates, the parametric equations $$x= 5 + \cos \theta$$ and $$y = \sin \theta$$ represent a circle $$\Gamma_1.$$ In Cartesian coordinates, there is a circle $$\Gamma_2$$ that is externally tangent to $$\Gamma_1,$$ tangent to the $$y$$-axis, and centered 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 $$r=14 \cos \theta\ (0 \leq \theta < 2\pi)?$$

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