Expert: Paul Klarreich - 2/18/2009

Question
A line passes through point (2, 2). Find the equation of the line if the length of the line segment intercepted by the coordinate axes is the side of a square whose are is 5 square units.  

Answer
Questioner:   lawrence
Category:  Advanced Math
Private:  No
 
Subject:  equation
Question:  A line passes through point (2, 2). Find the equation of the line if the length of the line segment intercepted by the coordinate axes is the side of a square whose are is 5 square units.
........................
If the equation is:

y - 2 = m(x - 2)

Then if x = 0, the y-int is:

y = 2 - 2m

And if y = 0, the x-int is obtained by

-2 = m(x - 2)
-2 = mx - 2m
x - 2 = -2/m
x = 2 - 2/m

So we have these facts relating x,y,m:

x = 2 - 2/m
y = 2 - 2m


But  x^2 + y^2 = 5, you said.

So:

(2 - 2/m)^2 + (2 - 2m)^2 = 5

4 - 8/m + 4/m^2 + 4 - 8m + 4m^2 = 5

3 - 8/m + 4/m^2 - 8m + 4m^2 = 0

Multiply by m^2 to get a polynomial:

3m^2 - 8m + 4 - 8m^3 + 4m^4 = 0

A quick and dirty graph suggests that:

Solutions are m = 2 or m = 1/2

From which you will get your x and y.
...............................

I will postpone your other questions.