find the equations of the normals to the curve xy=4 which are parellel to the line 4x-y-2=0
find the equations of the normals to the curve xy=4 which are parellel to the line 4x-y-2=0
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To find the equations of the normals to the curve xy = 4, which are parallel to the line 4x - y - 2 = 0, we can follow these steps:
1. Write the equation of the curve in terms of y: xy = 4
Divide both sides by x: y = 4/x
2. Differentiate the equation with respect to x to find the derivative of the curve:
dy/dx = -4/x^2
3. The slope of the line 4x - y - 2 = 0 is found by rearranging it into the slope-intercept form:
y = 4x - 2
The slope of this line is 4.
4. Since the normals to the curve are parallel to the line, the slope of the normals will also be 4.
5. Use the derivative of the curve to find the slope of the normals and equate it to 4 to get an equation:
-4/x^2 = 4
Multiply both sides by x^2: -4 = 4x^2
Divide both sides by 4: x^2 = -1
6. Notice that there are no real solutions for x^2 = -1, which means there are no parallel normals to the curve.
Therefore, there are no equations of the normals to the curve xy = 4 that are parallel to the line 4x - y - 2 = 0.