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.

Guest May 13, 2020

#1**+1 **

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

CPhill May 13, 2020