Find the normal equation to the curve LaTeX: xy\:+\:2x\:-\:y\:=\:0xy+2x−y=0 that are parallel to the line -4x - 2y = 10.
To find the normal equation to the curve that is parallel to the line -4x - 2y = 10, we first need to find the slope of the line.
The given line -4x - 2y = 10 can be rearranged into slope-intercept form (y = mx + b) by isolating y:
-4x - 2y = 10
-2y = 4x + 10
y = (-4/2)x - 5
y = -2x - 5
From this equation, we can see that the slope (m) of the line is -2.
The normal to a curve is perpendicular to the tangent of the curve at any given point. Since the desired normal is parallel to the given line, the slope of the desired normal must also be -2.
The equation of a line with slope -2 and passing through a given point (x₁, y₁) can be found using the point-slope form:
y - y₁ = m(x - x₁)
Now, we need to find a point on the curve that satisfies this condition.
The given curve equation is xy + 2x - y = 0. To find a point on this curve, we can choose a value for x and solve for y. Let's choose x = 0:
(0)y + 2(0) - y = 0
- y = 0
y = 0
So, the point (0, 0) lies on the curve.
Now, we can substitute the point (0, 0) and the slope (-2) into the point-slope form to find the equation of the desired normal:
y - y₁ = m(x - x₁)
y - 0 = -2(x - 0)
y = -2x
Thus, the normal equation to the curve xy + 2x - y = 0 that is parallel to the line -4x - 2y = 10 is y = -2x.