# The line that is normal to the curve x^2=2xy-3y^2=0 at(1,1) intersects the curve at what other point?

We have x2=2xy - 3y2 = 0
Are there supposed to be 2 equal signs in this expression or is it
x2 + 2xy - 3y2 = 0 ?
I'll suppose it's the second one.
You need to differentiate this implicitly to find dy/dx. Then find an equation for the normal line and set it equal to the curve.
The implicit derivative looks like
2x + 2x*dy/dx + 2y -6y*dy/dx = 0
Solve for dy/dx and use the negative reciprocal of dy/dx at the point (1,1) as the slope of the normal.
I'm not sure if you know what this second degree curve is, but it's an ellipse. The normal line at any point should intersect the ellipse in two points.

x2 + 2xy = 3y2
This is not an ellipse, but a pair of lines that pass through the origin, at least according to what I got in my graphics application.
Differentitate this to get
2x + 2x*dy/dx + 2y = 6y*dy/dx so
2(x+y) = 2*dy/dx(3y -x) and
dy/dx = (x+y)/(3y -x)
At (1,1) dy/dx = 2/2 = 1 so the normal has slope = -1
The equation of the normal is
y-1 = -1(x-1) or
y = -x + 2
Check the points (1,1) and (3,-1)

Sorry it is x^2+2xy-3y^2=0

How did you get the point (3,-1)? Thanks.

These are the lines y=x and y=(-1/3)x
The point (1,1) is on the line y=x with slope 1. The normal has slope -1 and goes through that point. It intersects the other line at (3,-1).

(2x-y)(3y+x)

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1. This might help: You already know how to find the equation for the normal line is y=-x+2. Plug this equation into the original equation of the curve in place of y:

x^2+(2x)(-x+2)-(3)(-x+2)^2=0
x^2-2x^2+4x-3x^2+12x-12=0
-4x^2+16x-12=0
(-4)(x^2-4x+3)=0
x^2-4x+3=0
(x-1)(x-3)=0

Now we know that x=1 and x=3 are the only two places where the graphs can cross. We already know when x=1 then y=1, so we only need to pay attention to x=3. Plug x=3 into the line for the normal curve to determine the y-value at that point:

y=-x+2
y=-3+2
y=-1

So we know the second point is (3, -1). I posted this response so belatedly because I had a tough time figuring this out from the responses I found (here and on other sites), so I hope this helps other students who are struggling with the concept.

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2. Awesome!

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