Normals parallel to a line?

Question

Find the normals to the curve xy+2x -y = 0 that are parallel to the line 2x + y = 0

this is from a quiz on implicit differentiation

Answer 1

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Normals ---> perpendicular

2x + y = 0, so y = -2x, and the slope = - 2

(xy) ' + ( 2x) ' - y ' = 0

x y ' + y + 2 - y ' = 0

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

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

now normal slope, M = neg reciprocal of the tangent slope to the curve...

M = - 1 / y ' = ( x - 1 ) / ( 2 + y ) = -2
now, from orig eqn... x ( y + 2 ) - y = 0, so x = y / ( y + 2 )

substitute in... - 2 = ( [ y / ( y + 2) ] - 1 ) / ( y + 2 )
simplifying... - 2 = [ y - ( y + 2 ) ] / ( y + 2 )^2
-2 = -2 / ( y + 2)^2

thus y + 2 = 1, or y + 2 = -1
thus y = - 1 , or y = - 3
then for y = -1, x = -1 .... and y = - 3, x = 3

thus you have the points ( -1, - 1) and ( 3, - 3 ), with a slope of - 2 ..

apply the eqn of a line to get ... y = - 2x - 3, and y = -2x + 3

graphing ...the original eqn... y = -2x / ( x - 1 ), and the three lines... y = -2x, y = -2x - 3, y = -2x + 3, on a TI Graphic,... you can see the lines are parallel, and each line we found is normal to the branch of the 'hyperbola' like curve, opposite to the points given... e.g., the normal thru pt. ( -1, -1 ) appears to be perpendicular to the branch opposite to the point ( - 1, - 1) .......weird huh ???!!!!

Answer 2

OK. Find dy/dx from xy+2x -y = 0 by implicitly differentiating it and factoring out the dy/dx. Now you'll have dy/dx = f'(x,y) and the parallel to the line has slope of -2 so you're looking for dy/dx = 1/2. At this point it's pretty much plug 'n play to find (x, y) values to get dy/dx = 1/2 (so the normal is -2). Then just generate the equations for the normals in point slope form and you're done.

Doug

Answer 3

xy + 2x - y = 0
y(x - 1) = - 2x
y = - 2x/(x - 1)
Differentiating implicitly,
xdy + ydx + 2dx - dy = 0
dy(x - 1) = - dx(y + 2)
dy/dx = - (y + 2)/(x - 1)
Normals will have a slope
y' = (x - 1)/(y + 2)
To be parallel to
y = - 2x
(x - 1)/(y + 2) = -2
x - 1 = - 2y - 4
2y = - (x + 3)
y = - (x + 3)/2
or
x + 2y + 3 = 0

Answer 4

take the spinoff of the function. then, placed one million/2 in for dy/dx and remedy. spinoff x(dy/dx) + y + 2 - dy/dx = 0 dy/dx = -y - 2 _____ x - one million ok , now i'm caught, it form of feels as though there are a limiteless form of solutions.................poop i hate no longer assisting yet i think of i'm on the wonderful course............