Given x+xy+2y^2=6 find the coordinates of all other points on this curve with slope equal to -1/3. Please show work.
To find the coordinates of all other points on the given curve with a slope equal to -1/3, we need to find the derivative of the equation and solve for the values of x and y.
First, let's rearrange the equation to isolate y:
x + xy + 2y^2 = 6
2y^2 + xy + x = 6
Now, to find the derivative, we differentiate each term with respect to x:
d/dx (2y^2) + d/dx (xy) + d/dx (x) = d/dx (6)
4y(dy/dx) + y + x(dy/dx) + 1 = 0 [Using the chain rule for the second term]
Now, we can simplify the equation:
(4y + x) (dy/dx) = - (y + 1)
Next, we want the slope to be equal to -1/3. So, let's set dy/dx = -1/3 and solve for the values of x and y:
(4y + x) (-1/3) = - (y + 1)
Multiply both sides by -3 to eliminate the fraction:
(4y + x) = 3(y + 1)
Expand the equation:
4y + x = 3y + 3
Simplify:
4y - 3y = 3 - x
y = 3 - x
Now we have an expression for y in terms of x.
To find all other points on the curve, we substitute this relation back into the original equation:
x + x(3 - x) + 2(3 - x)^2 = 6
x + 3x - x^2 + 2(9 - 6x + x^2) = 6
x + 3x - x^2 + 18 - 12x + 2x^2 = 6
Rearrange the terms:
3x - x^2 - 12x + 2x^2 + x - 18 + 6 = 0
Combine like terms:
x^2 - 8x - 12 = 0
Now we have a quadratic equation that we can solve.
Using factoring or the quadratic formula, we find that:
(x - 6)(x + 2) = 0
So, x = 6 or x = -2.
Substituting these values back into the expression for y, we get:
For x = 6: y = 3 - x = 3 - 6 = -3
For x = -2: y = 3 - x = 3 - (-2) = 5
Therefore, the other points on the curve with a slope of -1/3 are (6, -3) and (-2, 5).