there are two circles that go through (9,2). Each one is tangent to both coordinate axes.

Question

Find the center and the radius for each circle. Start by drawing a clear diagram.

Answer 1

 

In order for both circles to be tangent to both the x and the y axis, their centers have to be the same distance from both the x and y axis.  Thus the center of both circles must lie on the line y = x.

 

   Starting with the equation of a circle whose x and y coordinates are the same, we know that the radius of the circle is equal to the value of the x or y coordinate of the center ... (either one, because the (x, y) coordinates have the same value.

 

   We start with this equation:

 

   (x - h)^2 + (y - k)^2 = r^2

 

   For reasons stated earlier, ...

      h and k are identical, so we can use h for both values

      the radius is h

 

   Substituting in these values, as well as (9,2) fpr x and y ... since the circles both go through this point

   We end up with this equation:

 

    (9 - h)^2 + (2 - h)^2 = h^2

 

    SIMPLIFY:

 

    (9 - h)(9 - h) + (2 - h)(2 - h) = h^2

    81 - 18h + h^2 + 4 = 4h + h^2 = h^2

    h^2 -22h + 85 = 0

 

    FACTOR and use zero product property

 

    (h - 5)(h - 17) = 0

 

    Thus h = 5 or h = 17

 

    So the 2 equations for the 2 circles are:

 

      (x - 5)^2 + (y - 5)^2 = 5^2

 

      (x - 17)^2 + (y - 17)^2 = 17^2