Are the Curves 2x^2 + y^2 = 3 and x = y^2 Orthogonal?

[fuzzwolfie]

Homework Statement

Two curves are said to be orthogonal if their derivatives are opposite reciprocals at the point where the two curves intersect. Are 2x^2 + y^2 =3 and x= y^2 orthogonal?

Homework Equations

I'm not entirely sure what to put here, but I think one relevant thing is to say that the two curves will intersect if they have slopes that are perpendicular to each other. (so m2= -1/m1)

The Attempt at a Solution

Ok- so from what I can piece together, the first thing would be to find out if the two equations would have perpendicular slopes. So I would have to differentiate both equations to find their slopes.

so- 4x+ 2y= 0
and
1=2y

From here though, I am lost. I think y=.5 and x= -.25 and that m1=0 and m2=1. So my hunch is that these lines are not orthogonal. However, I am not sure if this is the correct thing to do. I have searched throughout my book and have not found any example problems worded like this. And related problems I found on the internet didn't make much sense to me.

First post here, so I hope I followed the form correctly. Any help or explanations of how to approach this problem would be very appreciated. Thank you for reading!

 

[Char. Limit]

Welcome to PF! :)

Well, the first thing I would do is actually find out if the two functions intersect, and if so, where?

 

[fuzzwolfie]

Welcome to PF! :)

Well, the first thing I would do is actually find out if the two functions intersect, and if so, where?

Ok! And thank you for the welcome! =)

ok. So I'll need to set them equal to each other, right? So I'd have 2x^2 + y^2 -3= y^2 -x

Which can simplify to 2x^2 + x -3= 0

So...to solve this I'd need to use the quadratic formula, correct? Which would give me x= 1 and -3/2.

From there I tried plugging the answers in. If I choose x=1, both equations turn into 1= y^2 so is this the value I need?

 

[Clever-Name]

What does 1 = y^2 imply about what y equals? The coordinates of intersection will have 1 x value but 2 y values

 

[fuzzwolfie]

What does 1 = y^2 imply about what y equals? The coordinates of intersection will have 1 x value but 2 y values

I would think it means Y can equal 1 or -1 and either squared would equal one. making he equation true. (1=1) Is this right?

 

[Clever-Name]

Yep! So that gives you two points of intersection: (1,1) and (1,-1)

Now go back to your criteria for orthogonality using those points!

 

[fuzzwolfie]

Yep! So that gives you two points of intersection: (1,1) and (1,-1)

Now go back to your criteria for orthogonality using those points!

Ok! So going back to the original 2 equations, I now must differentiate them and plug in the given values to see if the slopes are perpendicular to each other. When i plugged in (1,1) I received the slopes 6 and 1.

I then plugged in (1, -1) and got the slopes 2 and -3. Based of this, I can say that the lines are not perpendicular and therefore not orthogonal...right? =)