Expert: Josh - 4/3/2010

Question
I formulated an equation while working on a design concerning refraction from a parabolic/spherical mirror. r=(x^2+y^2)/2y. To apply this in an excel worksheet with the known values available I need to rephrase the formula with y as the unknown value. Can you please assist.
Regards

Answer
Dear Herman,

Try turning this into a quadratic equation in terms of y.

r*2y = x^2 + y^2
y^2 - 2ry + x^2 = 0

Can use completing the square or quadratic formula [Given ay^2+by+c=0, y=(-b +/- sqrt(b^2-4ac))/(2a)] to solve for y. Are you ok with this?

------Clarification-------

Given r=(x^2+y^2)/2y, we multiply both sides by 2y to get 2ry=(x^2+y^2).
Subtracting 2ry from both sides, we get y^2 - 2ry + x^2 = 0.  ....[#1]
This is a quadratic equation in y (i.e., y is treated as the unknown variable in this equation).

In general, a quadratic equation may be described by a*y^2 + b*y + c = 0 ....[#2]
where a, b and c are arbitrary real values.

[#1] is exactly like this. Comparing coefficients term by term between [#1] and [#2]:
For the y^2 term: a=1
For the y term:   b=-2r
For the constant term: c = x^2.

The quadratic formula finds two possible solutions for  a*y^2 + b*y + c = 0.
The roots are given by y=[-b+sqrt(b^2-4ac)]/(2a) and y=[-b-sqrt(b^2-4ac)]/(2a).
I am not going to prove this, but it follows from a procedure called "completing the square". If you are interested, you can find out more by looking this up on a search engine. The algebra is a little messy.

Since we determined a=1, b=2r, c=x^2, the solution for y are described by
either y = [-b-sqrt(b^2-4ac)]/(2a) = [2r-sqrt(4r^2-4x^2)]/(2) = 2r - sqrt(r^2-x^2)
or     y = [-b+sqrt(b^2-4ac)]/(2a) = [2r+sqrt(4r^2-4x^2)]/(2) = 2r + sqrt(r^2-x^2).
[note: you have only found half the answer]

Example: If x=1.5, y=4, then r=2.28125.
Using y = 2r-sqrt(r^2-x^2), gives y=0.5625 (a feasible answer, but not necessarily what you started with.) Instead, y = 2r+sqrt(r^2-x^2) returns y=4.