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
http://www.jiskha.com/display.cgi?id=1268357108
To find the normals to the curve that are parallel to the line 2x + y = 0, we can use the concept of slope.
First, let's find the slope of the line 2x + y = 0. This equation is in the form y = -2x, which means the slope of this line is -2.
Now, let's find the slope of the curve xy + 2x - y = 0 by rearranging the equation and putting it in slope-intercept form (y = mx + b):
xy + 2x - y = 0
Rearranging terms:
xy - y = -2x
Factoring out y:
y(x - 1) = -2x
Dividing both sides by (x - 1):
y = -2x / (x - 1)
To find the slope of the curve, we can take the derivative of y with respect to x:
dy/dx = (d/dx) (-2x / (x - 1))
Using the quotient rule for differentiation, we get:
dy/dx = (-2*(x - 1) - (-2x))/(x - 1)^2
= (-2x + 2 + 2x)/(x - 1)^2
= 2/(x - 1)^2
Now we have the expression for the slope of the curve as a function of x.
To find the normals to the curve that are parallel to the line 2x + y = 0, we need to find the values of x for which the slope of the curve is equal to the slope of the line, which is -2.
Setting the slopes equal to each other, we have:
2/(x - 1)^2 = -2
Simplifying, we get:
1/(x - 1)^2 = -1
Taking the reciprocal of both sides, we have:
(x - 1)^2 = -1
Since the square of a real number is always positive, there are no real values of x that satisfy this equation. Therefore, there are no normals to the curve xy + 2x - y = 0 that are parallel to the line 2x + y = 0.
Please note that the equation of the line 2x + y = 0 is a special case where the slope is -2, but for any other line parallel to this one, the process would be the same.