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A circle is tangent to the positive x-axis at x=3. It passes through the distinct points (6,6)  and (p,p) What is the value of  (p)? Express your answer as a common fraction.

 May 13, 2020
 #1
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One point  on the circle  is  (3,0)   and  the other  is  (6,6)

 

The center  must be   (3, b)  since the circle is tangent to the x axis

 

Since  the  radial distance from (3,0)  to (3, b)   =  the  same radial distance from (6,6) to ( 3,b)

 

So....using the square of the distances  we  have that

 

b^2   = ( 6-3)^2  + (6 - b)^2       

 

b^2  = 9  + 36 - 12b + b^2

 

12b = 45

 

b = 45/12   =  15/4  =  the radius of the circle

 

So  we  equation of the circle  is

 

(x - 3)^2  + (y - 15/4)^2 = (15/4)^2

 

Since (p,p)  is on the circle we have that

 

(p - 3)^2  + ( p - 15/4)^2  =(15/4)^2

 

p^2  - 6p + 9 + p^2 - (15/2)p + (15/4)^2 = (15/4)^2

 

2p^2  - (6 + 15/2)p  + 9  =  0

 

2p^2  - (27/2)p + 9  = 0

 

4p^2 - 27p + 18  =  0  factor  as

 

(4p - 3) ( p - 6)  =  0

 

Setting both factors to 0 and so;ving for  p we get that  p = 6   (we already know this)

 

or

 

p = (3/4)

 

So  (p,p)  = (3/4 , 3/4)

 

See the graph here  : https://www.desmos.com/calculator/hasagqw5bw

 

cool cool cool

 May 13, 2020

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