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How do I find the shortest distance from a line given in polar coordinates and a point given in Cartesian coordinates?

For example, say that the line is given by the polar coordinates rho = 2 and theta = 30 degrees, while the point is given by the Cartesian coordinates x = 7 and y = 7. How would I find the shortest distance from the point to the line? Thanks!

EDIT:

The line I described above is the line which passes though the polar point given by rho = 2 and theta = 30 degrees and is also perpendicular to the the line from the origin to this point. This is how I was defining this line using only rho and theta. Sorry for the confusion. With this added note, can anyone help? Thanks!

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    Well, the line passing through $(x_0,y_0)$ that is perpendicular to $(x_0,y_0)$ has the equation $\langle (x,y)-(x_0,y_0),(x_0,y_0) \rangle = 0$. $(x,y)$ are points on the line, and in your case, $(x_0,y_0) = (\sqrt{\frac{3}{4}}, \sqrt{\frac{1}{4}})$. This gives the equation of a line, and there is a standard formula to find the distance from a point to a line, or you can solve it as an optimization problem...2012-11-03

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First, use $r$ and $\theta$ to find an equation for the line you are describing in slope-intercept form.

In particular, convert your given point from polar coordinates to its Cartesian coordinates $(x,y)$, using $x = r\cos\theta$ and $y = r\sin\theta$.

Note also that the line from the origin through this point has slope $y/x$.

Now, to find the line this point describes (as in your question) note that you have a point on the line, namely, $(x,y)$, and know the slope will be $-x/y$. A single point and the slope are enough to find an equation for the line. (I'll leave that to you.)

Now the question becomes: given a line in slope-intercept form and a point's Cartesian coordinates, how do we find the shortest distance between them?

This last question has a standard formula whose derivation I assume you are familiar with.

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    The formula for $y$ should be $y = r \sin\theta$.2013-04-11