How do I find the normals to the curve
xy+ 2x - y = 0 that are parallel to the line 2x + y = 0
To find the normals to the curve that are parallel to a given line, you need to find the gradient of the given line first. Then, you can use this gradient to determine the slope of the normals. Let's go step by step.
Step 1: Find the gradient of the given line.
The equation of the line is 2x + y = 0. To find the gradient, rearrange the equation in the form y = mx + c, where m represents the gradient:
y = -2x (subtract 2x from both sides)
The gradient of the given line is -2.
Step 2: Find the slope of the normals to the curve.
The equation of the curve is xy + 2x - y = 0. To find the slope of the normals, we need to find the derivative of the equation with respect to x.
Differentiating the equation implicitly:
(x)(dy/dx) + y + 2 - (dy/dx) = 0
Rearranging the terms:
dy/dx (x - 1) = -y - 2
dy/dx = (-y - 2) / (x - 1)
To find the slope of the normals, we take the negative reciprocal of the derivative:
slope of the normals = -1 / dy/dx
slope of the normals = -(x - 1) / (y + 2)
Step 3: Equate the slope of the normals to -2 (the gradient of the given line).
-(x - 1) / (y + 2) = -2
Cross-multiply to get rid of the fraction:
-(x - 1) = -2(y + 2)
Expand and simplify:
-x + 1 = -2y - 4
-x + 2y = -5
2y = x - 5
y = (1/2)x - 5/2
The equation y = (1/2)x - 5/2 represents all the normals to the curve that are parallel to the line 2x + y = 0.