My brother and I have been working on this problem for the better part of the day. I think we keep overlooking something. Please help us with this problem.
Normals parallel to a line: Find the normals to the curve xy+2x-y=0 that are parallel to the line 2x+y=0.
Thank you in advance for your responces.
Ok, I see what you did. I kept differentiating implicity and I must have been overlooking a mistake (even though it should have worked too). But I came up with y=-2x+3 , y=-2x-3 as the equations of my normals. Everything looks good when I graph it too, so I think this is right. Thank you very much for your quick responce and your help.
There's a small mistake (edited in red - dang it, the very first line. Topsquawk was obviously thinking about Dora at the time .... lol!!). There's obviously a flow-on effect but the corrections are simple.
You should get the x-coordinates of the required points on the given curve are x = 3 and x = -1.
Edit: Reckoner, you should have posted your reply!
Obviously there's a bit more work if you use implicit differentiation (and therefore don't make y the subject). Anyway, since you mentioned it:
Implicit differentiation:
.
Therefore you require
..... (1)
Since the points on the curve that the required normals pass through obviously lie on the curve, you also require
...... (2)
Solve equations (1) and (2) simultaneously:
x = 3 and y = -3 or x = -1 and y = -1.
Equation of normal 1: m = -2 and passes through the point (3, -3). .
Equation of normal 2: m = -2 and passes through the point (-1, -1). .