Thread: Line intersecting implicit function

Hi, I have to find the co-ordinates where the line y=-x+1 intersects f(x,y) = x^2 + y^2 -xy - y -1

I can do this with explicit functions by making them equal, but it doesn't work with this.

The actual question asks for the co-ordinates where the tangent to f(x,y), is at 45 deg to the horizontal. So I found the gradient function which = 1 (that got me the linear equation), now i'm stuck.

Not sure if this should be in calc. section, so sorry if I got it wrong.
Thanks, Rich.

Hello hciR

Originally Posted by hciR

Hi, I have to find the co-ordinates where the line y=-x+1 intersects f(x,y) = x^2 + y^2 -xy - y -1

I can do this with explicit functions by making them equal, but it doesn't work with this.

The actual question asks for the co-ordinates where the tangent to f(x,y), is at 45 deg to the horizontal. So I found the gradient function which = 1 (that got me the linear equation), now i'm stuck.

Not sure if this should be in calc. section, so sorry if I got it wrong.
Thanks, Rich.

I assume that the curve is , in which case I agree that when .

Just substitute into to eliminate , and solve the resulting quadratic in . I make the answer

Then find the corresponding values of .

Grandad

The line y = -x + 1 lies in the x-y plane, or where f(x,y) = 0.

So sub that in:
0 = x^2 + y^2 - xy -y -1,
0 = x^2 + (-x+1)^2 -x(-x+1) -(-x+1) -1

Now solve for x.

EDIT - I got:

Oh ok, i was going about it the wrong way thanks.

As an aside:- I managed to get those co-ordinates for x earlier in the question, where it asked for the stationary points. Is it just the nature of the shape that the x co-ordinates are the same, when the gradient is 0 and 45 deg?

Also, when working out the S.P I got four possible Y values, and only knew which ones to use by plotting the graph, is there another mathematical way of doing it?
Thanks.