Solve the following equation so that it can be plotted on a graphing calculator.
4x^2 - 2xy + 3y^2 =2. I know it should be able able to use the quadratic
formula but every way I try, I do not get the right answer. The question came
from the TeXeS Review manual. There is no ISBN.
4x² - 2xy + 3y² = 2
Rearrange it in descending order of powers of y with 0 on the right:
3y² - 2xy + 4x² - 2 = 0
Write it this way, to see what A, B, and C are.
(3)y² + (-2x)y + (4x²-2) = 0
Then for the quadratic formula,
A = (3), B = (-2x), C = (4x²-2)
________
-B ± ÖB² - 4AC
y = —————————————————
2A
____________________
-(-2x) ± Ö(-2x)² - 4(3)(4x²-2)
y = —————————————————————————————————
2(3)
You can put that in the TI-83 just like this without simplifying further:
Y1 = (-(-2X) + Ö((-2X)² - 4(3)(4X²-2)))/(2(3))
Y2 = (-(-2X) - Ö((-2X)² - 4(3)(4X²-2)))/(2(3))
use window Xmin = -1, Xmax = 1, Xscl = 1, Ymin = -1, Ymax = 1, Yscl=1, Xres=1
You get a slanted ellipse
or you can simplify it further first
____________________
-(-2x) ± Ö(-2x)² - 4(3)(4x²-2)
y = —————————————————————————————————
2(3)
_______________
2x ± Ö4x² - 12(4x²-2)
y = ———————————————————————
6
_______________
2x ± Ö4x² - 48x² + 24
y = ———————————————————————
6
___________
2x ± Ö-44x² + 24
y = ———————————————————
6
___________
2x ± Ö4(-11x² + 6
y = ———————————————————
6
___________
2x ± 2Ö(-11x² + 6
y = ———————————————————
6
__________
2[x ± Ö-11x² + 6
y = ———————————————————
6
Cancel the 2 into the 6
_________
x ± Ö-11x² + 6
y = —————————————————
3
________
x ± Ö6 - 11x²
y = ———————————————
3
Then you can enter it in the TI-83 as
Y1 = (X + Ö(6 - 11X²))/3
Y2 = (X - Ö(6 - 11X²))/3
Either way you get this slanted ellipse.
Edwin