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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!


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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You've described a point in polar coordinates. A line through the origin would not specify rho, for example. – Muphrid Nov 3 '12 at 21:49
I'm not sure I understand. How is my (x, y) coordinates for the point in polar coordinates? Also, I know that a line through the origin would not specify a rho, but the example I'm giving is not one with a line through the origin. – Jenny Shoars Nov 3 '12 at 22:00
As @Muphrid noted, if you are in $\mathbb{R}^2$, then specifying the radius and angle uniquely defines a single point (you have $\rho = 2, \theta = \frac{\pi}{6}$, or $(x,y) = (\sqrt{\frac{3}{4}},\sqrt{\frac{1}{4}})$). – copper.hat Nov 3 '12 at 22:11
Ah, now I see what you're saying. It also defines a line though where the point given by rho = 2, theta = 30 degrees is the closest point of that line to the origin. In other words, the line is defined as passing through this point with the line from the origin to the point being perpendicular to the line. Make sense? – Jenny Shoars Nov 3 '12 at 23:19
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... – copper.hat Nov 3 '12 at 23:46
up vote 1 down vote accepted

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 $y = r \sin\theta$. – liborw Apr 11 '13 at 14:09

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