In Exercises 45–52, use your answers from Exercises 41–44 and ...

Question

In Exercises 45–52, use your answers from Exercises 41–44 and the parametric equations given in Exercises 41–44 to find a set of parametric equations for the conic section or the line.

Hyperbola: Vertices: (4,0) and (−4,0); Foci: (6,0) and (−6,0)

Accepted Answer

Welcome back everyone. In this problem, we want to determine the set of parametric equations for the hyperbola with vertices at 50 and the negative 50 and four at 80 and negative 80 A says X equals 56 T and Y is root 3910 TB says they are root 39 6 T and Y equals 510 C says it's eight ST and five to T and D says it's five to T and route 39 ST respectively. Now, what do we know about the parametric equations for our high parabola? Well, recall OK that for our parametric equations, X is going to be equal to a se OK? And Y OK is equal to B 20. OK. So now what this means then if we can find the values of A and B, we will be able to get the parametric equations. Now, what do we already know? Well, for starters, we know that for our higher parabola, it has vertices at 50 and negative 50. So that means then that the distance between the vertices, OK, the distance between vertices, OK, which would be two A, OK. Is going to be equal to the distance from five to negative five, which is 10. And if two A is 10, that means that A equals five. So we have the constant that we need or we have the value we need rather for the X equation. How about B? Well, here we're told that the four is at 80 and the negative 80. And that gives us the value of C or we can use that to find the value of C sorry. But not B however, we will be able to make use of that value to eventually help us solve for B because no, if we take that distance, OK. So the distance between the four chi, OK? And that distance is two C, it's gonna go from eight to negative eight. So that's 16, then that means C equals eight. Now, since C squared, OK is equal to A squared plus B squared, we can substitute our values for C and A to help us solve for B because no, what this tells us then is that eight squared is gonna be equal to five squared plus B squared. OK. If we solve here, eight squared equals 64 5 squared equals 25. And if we subtract 25 from both sides, B squared equals 64 minus 25 OK, which is equal to 39. And if B squared equals 39 OK, then that means B equals the square root of 39. Now that we have our values for A and B, this tells us, then that X is going to be equal to 56 te and Y is going to be equal to the square root of 39 multiplied by the tangent of T. If we look back at our answer choices, that means A is the correct answer. Thanks for watching everyone. I hope this video helped.

In Exercises 45–52, use your answers from Exercises 41–44 and the parametric equations given in Exercises 41–44 to find a set of parametric equations for the conic section or the line.

Hyperbola: Vertices: (4,0) and (−4,0); Foci: (6,0) and (−6,0)

Key Concepts

Parametric Equations of Conic Sections

Properties of a Hyperbola

Relationship Between Vertices, Foci, and Parameters a, b, c

Watch next