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**s_ingram** Hi guys,

I am struggling with this polar equation problem:

The polar equation of a circle is $\displaystyle r = 4\cos\theta$ and two points $\displaystyle P(4,0)$ and $\displaystyle Q(2\sqrt{2}, \frac{\pi}{4})$ are found where the circle intersects the line $\displaystyle r = 2\sqrt{2}\sec(\frac{\pi}{4}-\theta)$. Calculating P and Q was the first part of the problem. Now find the equations of the two half lines from the origin which are tangents to the circle which has PQ as diameter.

Maybe there is a way of doing this working directly with polar coordinates but I switch to cartesian and then convert back at the end. Perhaps this is not the best method - in which case I am open to suggestions.

In cartesian coordinates P is (4,0) and Q is (2,2) and the second circle with PQ as diameter can be seen to have radius $\displaystyle \sqrt 2$ and to pass through R(2,0) so I calculate it's equation to be $\displaystyle x^2 + y^2 -6x -2y + 8 = 0$. See attachment. But I can't see a way to calculate the tangents from the origin! I don't know which points on the second circle will have tangents passing through the origin.

I tried implicit differentition to get a general gradient but I couldn't see how to use it. Can anyone help?