Question

Find the equation to the normal curve xy+2x-y=0 which are parallel to line 2x+y=0​

Answer 1

If a normal to a curve is parallel to 2x + y = 0 then these two lines have the same slope:

y = -2x

slope = -2

Since the normal line at a point is perpendicular to the tangent line at that point, the slope of the tangent line is 1/2

xy+2x-y=0 => implicit differentiation

xy' + y + 2 - y' = 0

y'(x - 1) = -y-2

y' = (-y-2)/(x-1) => equate to slope of the tangent(1/2)

1/2 = (-y-2)/(x-1)

y = (-1/2)x -3/2 => the equation of the tangent, solve as a system with the curve to get the intersections:

xy + 2x - y = 0

x = - 1 , 3

y = -1 , -3

m = -2 , (-1 , -1)

y - y1 = m(x - x1)

y + 1 = -2(x + 1)

y = -2x - 3 => equation of the normal at(-1 , -1)

y + 3 = -2(x - 3)

y = -2x + 3 => equation of the normal at (3 , -3)