Find all points on the curve dy/dx = (-2x-y)/(x+2y) where the tangent line is parallel to the line y= -2x
To find the points on the curve where the tangent line is parallel to the line y = -2x, we need to find the values of x and y that satisfy two conditions: the given slope dy/dx = (-2x - y)/(x + 2y) and the slope of the line y = -2x.
The slope of the line y = -2x is -2.
To find the points on the curve where the tangent line has the same slope, we can set the slope of the curve equal to -2 and solve for x and y.
(-2x - y)/(x + 2y) = -2
Multiply both sides of the equation by (x + 2y) to eliminate the denominator:
-2x - y = -2(x + 2y)
Distribute -2 on the right side:
-2x - y = -2x - 4y
Rearrange the equation to isolate y:
-2y + 4y = -2x + y
2y = -x + y
y = -x
Now substitute this value of y into the equation dy/dx = (-2x - y)/(x + 2y):
dy/dx = (-2x - (-x))/(x + 2(-x))
dy/dx = (-2x + x)/(x - 2x)
dy/dx = -x/(-x)
dy/dx = 1
So, the slope of the tangent line is 1 when y = -x.
Therefore, the points on the curve where the tangent line is parallel to the line y = -2x are the points where y = -x. These points can be represented as (x, -x), where x is any real number.