Math Is Fun Forum

Beriz

Member

Registered: 2007-08-25

Posts: 1

Equation of a circle

Can someone please help me with this question? Find an equation of a circle with center at origin that passes through (10,8)

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Equation of a circle

My first thought was:  That's not a circle, silly, because it would be (9,9) or (10,10).
But now let me draw picture and get back to you.
Remember  r*r = x*x + y*y, where x = 10 and y = 8.
So 100 + 64 = r * r, where * means multiply.
Calculator time: 12.80624847 is r, thereabout.
Or factor 164 into half.  82 41...  41 * 2 * 2 is 164
2√41 = r = 12.80624847 thereabout.
It's like pythagoreans rule for right triangles.
The hypotenuse is the radius I guess.
Well I never did draw a picture, but I think if you divide 12.80624847 by 1.4142136
then you will get the double pair coordinates at 45 degree angle from origin.
1.4142136 is cosine and sine of 45 degrees, and the square root of 2.
(9.055385138 , 9.055385138) is a coordinate on the circle I guess, or very close to it.
Oh, and that's the square root of 82, by the way.
Now (10,8) is at an angle of maybe around 39 degrees from the origin,
like to the right mostly, but upward too.
39 is my guess.  If you want the real answer, then do rise over run and tangent that.
Like y=8 divided by x=10 to the right,  8 up and 10 right divided is 0.8 or 80%.
Now tangent of 80% is what.  Use calculator or a look-up table.
Wait, Use inverse tangent, sorry, not tangent.
So 2nd function to make inverse tangent on calculator of 0.8 and get 38.65980826 degrees.
Wow, I guessed pretty close, huh?

Last edited by John E. Franklin (2007-08-25 06:08:50)

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bossk171

Member

Registered: 2007-07-16

Posts: 305

Re: Equation of a circle

Remember:

Now it's centered at the origin (0,0) so we know that h and k are both 0.

That in mind, out new simplified equation is:

At some point it passes through (10, 8) (so says you) so we know that:

must be true at some point. so:

Now that we know h, k, and r^2 we can say that:

PS: If you have access to a Graphing Calc graph

and

to check your answer.

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Equation of a circle

Problem answered.  Now it's time to abstract!

(i) Find a function f(x, y) that takes a point (x, y) as input, and gives you the radius of a circle centered at the origin that travels through that point.

(ii) Redefine the function to take a single value that can be computed using x and y.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: Equation of a circle

(i) isn't that just f(x,y) = sqrt(x^2 + y^2) ?

i'm not sure what you're saying in (ii).

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A logarithm is just a misspelled algorithm.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Equation of a circle

Yes, I'm confused by (ii).

It seems like he wants some function g(z) = r, where z is made by combining x and y in some way. But if that's the case then you could just make z by the function in (i) and say that g(z) = z.

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mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: Equation of a circle

hehehe!

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A logarithm is just a misspelled algorithm.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Equation of a circle

Yea, part 2 I added in at the last second, didn't really think it through though.  Anyways, it was just an exercise in abstractions, not exactly meant to be difficult.  Now do it for R^n.  Then once you finish that, do it for metric spaces.  Then topological spaces.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."