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**mathstudent2000****Member**- Registered: 2013-07-26
- Posts: 79

1. For some positive real number r, the line x + y = r is tangent to the circle x^2 + y^2 = r. Find r.

2. Find the center of the circle passing through the points (-1,0), (1,0), and (3,1). Express your answer in the form "(a,b)."

3. A line with slope 3 is 2 units away from the origin. Find the area of the triangle formed by this line and the coordinate axes.

4. Find the maximum value of y/x over all real numbers x and y that satisfy (x - 3)^2 + (y - 3)^2 = 6.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,694

Hi mathstudent2000;

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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**mathstudent2000****Member**- Registered: 2013-07-26
- Posts: 79

For no. 3 its not 20/3, but 6 because i tried it yesterday and got it correct

Genius is one percent inspiration and ninety-nine percent perspiration

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,694

Who said it was 20 / 3?

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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**mathstudent2000****Member**- Registered: 2013-07-26
- Posts: 79

sorry, i think i saw another problem's answer

Genius is one percent inspiration and ninety-nine percent perspiration

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,694

Hi;

We are all done here?

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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**champion999****Member**- Registered: 2015-07-01
- Posts: 2

The correct answer for (3) was actually 20/3.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,694

Hi;

That is the answer I would have used also but sometimes you have to guess at what the OP wants.

For no. 3 its not 20/3, but 6 because i tried it yesterday and got it correct.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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**RandomPieKevin****Member**- Registered: 2015-07-02
- Posts: 29

Can someone explain how to get the answer for Q4? Thanks.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,694

I answered this question and now I am wondering how I did it?! What does this x/y mean?

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,016

You can do it by finding the tangents from (0,0) to the circle.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,694

You might use Lagrangian Multipliers too.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,659

4. Find the maximum value of y/x over all real numbers x and y that satisfy (x - 3)^2 + (y - 3)^2 = 6.

That locus is a circle, centre (3,3) and radius root 6.

Pick any point on the circle and join to the origin. The gradient of the line will be the value of y/x .

To maximise this value move the point around the circle until the line makes a tangent to the circle, at approximately (0.59, 3.38).

You can calculate the exact point using coordinate geometry.

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**math9maniac****Member**- From: Tema
- Registered: 2015-03-30
- Posts: 373

For no. 2, you put the values of x and y into the general equation of the circle, for all three points. You obtain 3 equations with three variables. Solve simultaneously. The center of the circle is (-g, -f).

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,659

As already stated by two people you need to make a tangent to the circle from the origin. One will have the maximum gradient; the other has the minimum.

Let the point where the max tangent hits the circle be P, and the centre of the circle be C, origin O.

Then OC^2 = CP^2 + PO^2 => OP = root 12.

So P is on a circle x^2 + y^2 = 12.

Put this with the other circle to get x + y = 4. Substitute y = 4 - x into the first circle and solve for x.

You'll get two answers. One is the max and one the min.

Find y and compute y/x

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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