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#1 2013-08-15 12:51:12

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

geometry problems

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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#2 2013-08-15 18:52:31

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 96,622

Re: geometry problems

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.

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#3 2013-08-16 05:12:01

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

Re: geometry problems

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


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#4 2013-08-16 05:26:15

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 96,622

Re: geometry problems

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.

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#5 2013-08-17 04:42:47

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

Re: geometry problems

sorry, i think i saw another problem's answer


Genius is one percent inspiration and ninety-nine percent perspiration

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#6 2013-08-17 04:49:33

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 96,622

Re: geometry problems

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.

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#7 2015-07-10 11:58:16

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

Re: geometry problems

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

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#8 2015-07-10 12:59:58

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 96,622

Re: geometry problems

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.

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#9 2015-07-27 05:28:06

RandomPieKevin
Member
Registered: 2015-07-02
Posts: 28

Re: geometry problems

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

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#10 2015-07-27 06:56:09

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 96,622

Re: geometry problems

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.

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#11 2015-07-27 12:57:31

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 15,937

Re: geometry problems

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


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#12 2015-07-27 13:17:49

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 96,622

Re: geometry problems

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.

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#13 2015-07-27 19:59:25

bob bundy
Moderator
Registered: 2010-06-20
Posts: 7,056

Re: geometry problems

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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#14 2015-07-28 07:56:50

math9maniac
Member
From: Tema
Registered: 2015-03-30
Posts: 215

Re: geometry problems

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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#15 2015-07-29 02:40:59

RandomPieKevin
Member
Registered: 2015-07-02
Posts: 28

Re: geometry problems

Can someone explain how to do Q4: Find the maximum value of

over all real numbers
and
that satisfy
?

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#16 2015-07-29 02:58:07

bob bundy
Moderator
Registered: 2010-06-20
Posts: 7,056

Re: geometry problems

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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