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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.


Genius is one percent inspiration and ninety-nine percent perspiration

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

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 105,242

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.
No great discovery was ever made without a bold guess.

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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


Genius is one percent inspiration and ninety-nine percent perspiration

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

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 105,242

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.
No great discovery was ever made without a bold guess.

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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: 105,242

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.
No great discovery was ever made without a bold guess.

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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: 105,242

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.
No great discovery was ever made without a bold guess.

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

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

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: 105,242

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.
No great discovery was ever made without a bold guess.

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

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

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: 105,242

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.
No great discovery was ever made without a bold guess.

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

bob bundy
Moderator
Registered: 2010-06-20
Posts: 7,651

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: 372

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).


Only a friend tells you your face is dirty.

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#15 2015-07-29 02:58:07

bob bundy
Moderator
Registered: 2010-06-20
Posts: 7,651

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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