Question

Last question of the night!!!

A circle in the first quadrant with center on the curve y = 2x2 − 27 is tangent to the y-axis and the line 4x = 3y. The radius of the circle is m/n where m and
n are relatively prime positive integers. Find m + n.

Answer 1

y = 2x^2

Answer 2

xD

Answer 3

The center is on the curve 2x^2? Not on the curve 2x^2-27?

Answer 4

the center is on the curve y = 2x^2-27

Answer 5

i forgot to put the "^" in the original question xD

Answer 6

Ah. Here's my thought process...
For a given point on that curve, the radius of the circle will be DEFINED by the perpendicular distance to the y axis.
If that radius is the same as the nearest point on y = (4/3)x, then that's our circle.

Answer 7

For a given point (x,y) on the line, the distance to the y axis is just x.
The distance to the line y = (4/3)x is:
|4x -3y|/sqrt(25) = |4x-3y|/5
From here:

Answer 8

Setting those equal:
x = |4x-3y|/5

Now, y is a function of x. y = 2x^2 -27, so substituting in gives:
x = |4x -3(2x^2-27)|/5
=|-6x^2 +4x+81|/5

Answer 9

Solve for x.

Answer 10

however, the circle defined above which satisfies the conditions is Not in the 1st quandrant, the center would lie in the 4th quadrant.

Answer 11

Oh really?

Answer 12

Well damn...

Answer 13

wait nevermind i didn't look at the neg case of absolute value
it can also be in 1st quadrant
wow this problem is giving me fits for some reason

Answer 14

http://www.wolframalpha.com/input/?i=plot++y+%3D+%284%2F3%29x%2C+y+%3D+2x^2+-27+%2C+%28x-4.5%29^2%2B%28y-13.5%29^2+%3D+20.25%2C+from+x%3D0+to+9

radius = 4.5 = 9/2
m+n = 11