Write the hyperbola in standard form with the vertices at (1,1) and (5,1) and the foci at (-1,1) and (7,1)

Question

Write in standard form 

Answer 1

Vertices:   (1, 1) and (5, 1)

Foci:   (-1, 1) and (7, 1)

Notice that the vertices and the foci all have the same y-coordinate (that being 1) which means that the transverse axis is horizontal. The standard form for the equation of a hyperbola with a horizontal transverse axis is as follows:

          (x - h)²/a²  -  (y - k)²/b²  =  1

     where the center of the hyperbola is at the point (h, k), the vertices of the hyperbola are at the points (h+a, k) & (h-a, k), and the foci of the hyperbola are at the points (h+c, k) & (h-c, k)

Vertices:   (1, 1) = (h+a, k)   ==>   h + a = 1   ==>   h = 1 - a

                (5, 1) = (h-a, k)   ==>   h - a = 5   ==>   h = 5 + a

     1 - a = 5 + a   ==>     1 - a + a - 5 = 5 - 5 + a + a

                           ==>     -4 = 2a   ==>   -4/2 = 2a/2  

                           ==>   -2 = a     ==>   (-2)² = 4   ==>   a² = 4

Foci:   (-1, 1) = (h+c, k)   ==>   h + c = -1   ==>   h = -1 - c

          (7, 1) = (h-c, k)   ==>   h - c = 7   ==>   h = 7 + c

     -1 - c = 7 + c   ==>   -1 - 7 - c + c = 7 - 7 + c + c   

                            ==>   -8 = 2c   ==>   -8/2 = 2c/2  

                            ==>   -4 = c   ==>   (-4)² = 16   ==>   c² = 16

To find b, use the following equation:     a² + b² = c²  

          ==>   b² = c² - a²   ==>   b² = (-4)² - (-2)² = 16 - 4 = 12

          ==>   b² = 12   ==>   b = √12

Since the center is at (h, k) and all the y-coordinates are the same, this means that  k = 1

To find h, use one of the equations we solved for h and plug in the missing variable:

          h = 5 + a ,   a = -2   ==>   h = 5 + -2 = 5 - 2 = 3

            OR

          h = 7 + c ,   c = -4   ==>   h = 7 + -4 = 7 - 4 = 3

     Therefore,   h = 3

Given the standard form of the equation at the top of the page, we can plug in all missing variables we solved for:

               (x - 3)²/4  -  (y - 1)²/12  =  1

Answer 2

Madison,

This is essentially the same question I just answered you, except that the hyperbola is East-West opening , so its equation has the form
(x−h)²/a² - (y−k)²/b²= 1

Follow the same logic to figure out (h, k), a, c and b, you have your equation.

Best,

Heng