# SOLUTION: Find Standard Equation Of a Hyperbola: Given, Foci: (-8,0) and (8,0) Vertices: (-6,0) and (6,0).

Algebra ->  Algebra  -> Quadratic-relations-and-conic-sections -> SOLUTION: Find Standard Equation Of a Hyperbola: Given, Foci: (-8,0) and (8,0) Vertices: (-6,0) and (6,0).       Log On

 Ad: Algebrator™ solves your algebra problems and provides step-by-step explanations! Ad: Algebra Solved!™: algebra software solves algebra homework problems with step-by-step help!

 Algebra: Conic sections - ellipse, parabola, hyperbola Solvers Lessons Answers archive Quiz In Depth

 Click here to see ALL problems on Quadratic-relations-and-conic-sections Question 170281: Find Standard Equation Of a Hyperbola: Given, Foci: (-8,0) and (8,0) Vertices: (-6,0) and (6,0). Answer by jim_thompson5910(28593)   (Show Source): You can put this solution on YOUR website!Take note that ALL of the points given to you (both vertices and foci) all have a y-coordinate of 0. So this tells us that the hyperbola opens left and right like this: Take note that the distance from the center to either focus is 8 units. So let's call this distance "c" (ie ) Remember, the equation of any hyperbola opening left/right is So we need to find the values of h, k, a, and b Now let's find the midpoint of the line connecting the vertices. This midpoint is the center of the hyperbola x mid: Average the x-coordinates of the vertices: So the x-coordinate of the center is 0. This means that h = 0 ------------------- y mid: Average the y-coordinates of the vertices: So the y-coordinate of the center is 0. This means that k = 0 So the center is (0,0) which means that h=0 and k=0 ------------------ Now because the hyperbola opens left and right, this means that the vertices are (h+a,k) and (h-a,k). In other words, you add and subtract the value of "a" to the x-coordinate of the center to get the vertices. Since the value of "h" and "k" is 0, this means that the vertices become (0+a,0) and (0-a,0) then simplify to (a,0) and (-a,0) So this tells us that a=6 and -a=-6 which simply means that a=6 ------------------------------------------------------------------- Now it turns out that the value of "b" is closely connected to the values of "a" and "c". They are connected by the equation Plug in and Square 6 to get 36. Square 8 to get 64 Subtract 36 from both sides. Subtract Take the square root of both sides. Note: only the positive square root is considered (since a negative distance doesn't make sense) Simplify the square root. ========================================================== Recap: So we found the following: , (the x and y coordinates of the center), and Start with the general equation for a hyperbola (one that opens left/right) Plug in , , and Square 6 to get 36. Square to get Multiply Simplify ================================================================== Answer: So the equation of the hyperbola that has the foci (8,0) and (-8,0) along with the vertices (-6,0) and (6,0) is: