We're giving the equation for the track tricks curve where hell is greater than zero. And we want to show that it has the following properties that for all t the segment from C A T to T comma zero is tangent to the curve. It has a length? No. So you have our track drinks curve there and here I he produced the figure a little. So we have curve on a tangent coming off of that curve at some point, Tina and that tangent intersex, the X axis value and the length distance from this point at this point, is that so? First, we need to get the tangent of this track tricks function. And so the tangent rector is simply this. Okay, so we can use a trick. Identities, Just take the derivative of these hyperbolic metabolic functions and get this value here. So what we want to do is you want to say well, the line This line here has the slope of why component of the tangent divided by the ex comport Qianjin on That turns out to be the negative Um Cole Hyperbolic Oh, seeking one over the hyperbolic sign of tea not over and then we know that we need to take the slope times X minus X not It's not ISS. This guy evaluated Aquino and then we need to add on Why not? Is this guy evaluated? And you can figure out if we do some simplification and you some hyperbolic identities, you get that this thing is actually the quantity x 14 at minus x times the hyperbolic coast seeking of tangent of Tina over l. So that's what this this line is. So what we can see pretty clearly is that, um when x equals t not why you go zero So, in fact, meet cross zero at this point so that it's true and said I would lean the find the distance all right. And so we get this result. Obviously, when X is Tina, why is zero now? We want to find the distance. So we have two points. We have a point on the track tricks and our point on the x Acto y it on the X axis. And then we just need to figure out the distance between those two points and we have those two points. And if we go through, no. If we plug those plug that in. You get that. We need to take the difference. I think something is cancel out. When we get that. We need the magnitude of this record. Um, and we can using some hyperbolic identities, we can show that the magnitude that this isn't just out.