How can we show that the the center of a circle with radius 72 is (72,0)?

1 Answer
Morgan
Aug 7, 2017

See below.

Explanation:

In order to see why the radius of the circle is 72 with center (72,0), we'll have to manipulate the equation a bit. We have:

r=7cosθ

Let's multiply both sides by r:

r2=7rcosθ

Since, in the polar coordinate system, r2=x2+y2 and x=rcosθ, we see that we now have, in rectangular form:

x2+y2=7x

This is still the equation of a circle, just in a less-than-beautiful form which is also a bit difficult to interpret. Let's see if we can make it more useful.

Subtracting 7x from both sides:

x2−7x+y2=0

Let's treat this like x2−7x+0+y2=0 and see if we can complete the square. We take (72)2=494 and so now we have:

x2−7x+494−494+y2=0

⇒x2−7x+494+y2=494

Now we can make a perfect square

(x−72)2+y2=494

which looks much more like the equation of a circle that we're used to seeing! The general equation of a circle in rectangular form is:

(x−h)2+(y−k)2=r2

where r is the radius of the circle centered at (h,k)

We have:

(x−72)2+(y−0)2=(72)2

So we can now see that h=72,k=0, and r=72. This gives us a circle with radius 72 and center (72,0).

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