For the curve given by 4x^2 + y^2 = 48 + 2 * x * y, find the positive y-coordinate given that the x-coordinate is 2.
insert into the equation for x, two. Then solve for y.
16+y^2=48+4y
y^2-4y-32=0
(y-8)(y+4)=-
y=8, or -4 when x=2
To find the positive y-coordinate for a given x-coordinate of 2, we need to substitute x = 2 into the equation 4x^2 + y^2 = 48 + 2 * x * y and solve for y.
Step 1: Substitute x = 2 into the equation:
4(2)^2 + y^2 = 48 + 2 * 2 * y
Simplifying:
16 + y^2 = 48 + 4y
Step 2: Move all terms to one side of the equation:
y^2 - 4y - 32 = 0
Step 3: Factor the quadratic equation:
(y - 8)(y + 4) = 0
The equation is now factored, and we can solve for y:
Case 1:
y - 8 = 0
y = 8
Case 2:
y + 4 = 0
y = -4
Since we are looking for the positive y-coordinate, the solution is y = 8.
Therefore, for the curve given by 4x^2 + y^2 = 48 + 2 * x * y and x = 2, the positive y-coordinate is 8.