Find the normals to the curve
xy + 2x - y = 0 that are parallel to the line 2x + y = 0.
I can find the normal lines and the slope but I don't know what I'm suppose to do with the line 2x + y = 0
The question requires that among the family of normals to the curve, those parallel to the given line are required, that is where 2x+y=0 come in.
y=1x/2-4/3
To find the normals to the curve that are parallel to the line 2x + y = 0, we need to find the slopes of both the curve and the line.
First, let's find the slope of the given line. The equation of the line can be rearranged into the slope-intercept form y = mx + c, where m is the slope. In this case, the equation 2x + y = 0 can be written as y = -2x, so the slope of the line is -2.
Now, let's find the slope of the curve. The given equation xy + 2x - y = 0 can be rearranged to isolate y: y = -2x/(x - 1). To find the slope of the curve at any point, we can differentiate this equation with respect to x using the quotient rule. Taking the derivative, we get:
dy/dx = [-2(x - 1) - (-2x)] / (x - 1)^2
= (-2 + 2) / (x - 1)^2
= 0 / (x - 1)^2
= 0
Since the derivative is zero, it means that the slope of the curve at any point is zero. This implies that the curve is a horizontal line.
Since the given line has a slope of -2 and the curve is a horizontal line with a slope of zero, there are no points on the curve where the normal line can be parallel to the line 2x + y = 0. Therefore, there are no normals to the curve that are parallel to the given line.