Determining equation of vertical line tangent to curve
About this video
The video finds the vertical line tangent to the curve x^2 y + y^2 x = -2. A vertical tangent happens where dx/dy = 0, so you set the numerator of dx/dy (x^2 + 2xy) to zero, solve for y, and substitute into the original curve to get x = 2.
You'll be able to
- explain why a vertical tangent line has dx/dy = 0.
- find where a vertical tangent occurs by setting the numerator of dx/dy to zero with a non-zero denominator.
- substitute back into the original curve to find the equation of the vertical tangent line.
Key words
- vertical tangent line
- A tangent line that goes straight up and down, with equation x = some number.
- dx/dy
- The rate at which x changes with respect to y; it is the reciprocal of dy/dx.
- implicit curve
- A curve given by an equation in x and y that is not solved for y.
Transcript
- [Instructor] Consider the curve given by the equation x squared y plus y squared x is equal to negative two, it can be shown that the derivative of y with respect to X is all of this business. Write the equation of the vertical line that is tangent to the curve.
All right, this is interesting. Pause this video, have a go at that before we do it together. All right, so let's just visualize. I don't know exactly what this curve looks like, but let's just assume it's something like that and there's a tangent, there's a vertical line that is tangent to the curve at some point.
So what we wanna do is figure out the equation of this and the equation of a vertical line is gonna be x is equal to something. Now what do we know about a vertical line like this? Well, you could say that the slope is infinite. Or another way you could think about it is the change...
Just as we would say that for a horizontal line, if we had something like that, we would say that for a horizontal line the derivative of y with respect to x is equal to zero. Y is not changing with respect to x. For a vertical line, you don't see this a lot, but you could make the argument that for a vertical line the derivative of x with respect to y is equal to zero.
But we have here the derivative of y with respect to x. Well, we can just flip it. We could just say, all right, well then dx, dy is equal to x squared plus 2xy over negative 2xy minus y squared. Now, when is this gonna be equal to zero? This is gonna be equal to zero if the numerator equals zero and the denominator does not equal zero.
That also makes sense when we think about it from the derivative of y with respect to x approaching infinity or maybe getting to infinity, whatever that means. Because once again, you would want a situation in this case where the denominator is equal to zero and the numerator isn't equal to zero.
But in either case, we wanna figure out when x squared plus 2xy is equal to zero and this stuff down here is not equal to zero. And we also want that point or the set of points to be on the original curve. Remember, this is a tangent line so it has to go, that x value has to sit someplace on here.
So let's try to figure out when does this numerator equal zero? So x squared plus 2xy, this numerator or this denominator here on the left is equal to zero. And what I'm gonna do is I'm actually gonna solve for y here and then I'm gonna substitute it back into the original curve and then I can solve for this x.
So let's see if I'm solving for, if I wanna solve for y, I could say x squared is equal to negative 2xy and now I could divide both sides by negative 2x and let me just do that, negative 2x, negative 2x. These cancel out and if I flip both sides, I get y is equal to negative x over two, all right.
Now I can substitute that back into this original equation right over here. So every place I see a y, I'll just substitute it with that. And so I would get x squared times y, but now we know y is negative x over two. Negative x over two plus y squared. So that's negative x over two squared, which is the same thing.
If I square a negative it becomes a positive, it becomes x squared over four. All I did is I, that's y squared right over there, times x is equal to and did I write over, that's a negative two there, right? I wanna make sure I can still read what I originally wrote.
Actually, let me erase my writing here just to make sure. Yes, there was definitely a negative there. Okay, is equal to negative two. And so let's see, I get x to the third, negative x to third over two plus x to the third over four is equal to negative two. And since I have different denominators here, let me make them have the same denominator.
So if I want a denominator of four here, I multiply the denominator by two. So let me multiply the numerator by two as well. So negative 2x to third over four plus x to third over four is gonna be equal to negative x to third over four is equal to negative two.
Multiply both sides by four, I get negative x to the third is equal to negative eight, multiply or divide both sides by negative one and I am going to get x to the third is equal to eight or X is equal to two. So there you have it. We have the equation of the vertical line that is tangent to the curve.
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