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A circle C is tangent to the line y = -2, and also externally tangent to the circle x^2 + y^2 = 1.  Given that the center of circle C is O = (16,a), what is the value of a?

 Feb 20, 2020
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Let  the  distance  from   the center  of the  circle  to  the  line  y  = -2  be

 

sqrt  [ ( a + 2)]^2     this is the radius of the  circle

 

And the center  of  x^2 + y^2  =1   is  (0, 0)    and its radius  = 1

 

And  since circle  C  is tangent to  the circle  x^2  + y^2  =1

 

Then  the distance  from   ( 16,a)  to  this center  is  1 greater than the radius

 

So...we have this equation

 

sqrt [ (a  + 2)^2 ]+ 1   = sqrt  [16^2  +  a^2 ]

 

(a + 2)+  1   = sqrt (256 + a^2)

 

a + 3   = sqrt (256 + a^2)       square both sides

 

a^2  + 6a  + 9 =   a^2  + 256     subtract a^2  from both sides

 

6a + 9  =  256

 

6a  = 247

 

a  =  247 / 6

 

 

cool cool cool

 Feb 20, 2020

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