Hyperbolas
posted by Tabby .
Find the equation of the hyperbola whose vertices are at (1,5) and
(1,1) with a focus at (1,7). Please and thank you.

Since the vertices and focus are on the same vertical line (x = 1), this hyperbola has an equation in the form:
(y  k)^2/a^2  (x  h)^2/b^2 = 1
The center of the hyperbola is at the midpoint of the transverse axis:
h = [1 + (1)]/2 = 1
k = [5 + 1]/2 = 2
The distance between the center (1,2) and a vertex (1,5) is 3 units, so a = 3. The distance between the center (1,2) and a focus (1,7) is 5 units, so c = 5.
Using c^2 = a^2 + b^2, we have:
b^2 = c^2  a^2 = 25  9 = 16
Substituting 1 for h, 2 for k, 9 for a^2, and 16 for b^2:
[y  (2)]^2/9  [x  (1)]^2/16 = 1
Simplifying:
16(y + 2)^2  9(x + 1)^2 = 144
16(y^2 + 4y + 4)  9(x^2 + 2x + 1) = 144
16y^2 + 64y + 64  9x^2  18x  9 = 144
16y^2 + 64y + 64  9x^2  18x  9  144 = 0
16y^2 + 64y  9x^2  18x  89 = 0
If I haven't missed anything, this should give you an idea of how to do these kinds of problems.
I hope this helps.
Respond to this Question
Similar Questions

math
the vertices are at (2,1) and (2,7) and focus is at (2,*) write the equation of the hyperbola that meets each set of conditions. Is the asterisk supposed to be an 8? 
PreCalculus
Find the equation of the hyperbola whose vertices are at (1,5) and (1,1) with a focus at (1,7). So far I have the center at (1,2) and part of the equation is (y+2)^2  (x+1)^2 but do not know how to figure a^2, b^2, or c^2. … 
Algebra 2
Choose the equation that best represents an ellipse for the given foci and covertices. 1. foci (+2, 0) covertices (0, +4) 2. foci (+3, 0) covertices (0,+6) 3. foci (0, +3) covertices (+5, 0) Choose the equation for the hyperbola … 
pre cal
What is the center of the conic whose equation is x^2 + 2y^2  6x + 8y = 0 2.Which one of the following equations represents a hyperbola? 
Math Help PLEASE!
Find the equation of the hyperbola whose vertices are at (1,5) and (1,1) with a focus at (1,7). Please and thank you. 
algebra
a hyperbola has vertices (+5,0) and one focus (6,0) what is the standard form equation of the hyperbola. 
algebra
Can someone please help... a hyperbola has vertices (+5,0) and one focus (6,0) what is the standard form equation of the hyperbola. 
hyperbolas
Find the equation of the hyperbola that has a center at (8,1), a focus at (3,1), and a vertex at (4,1). 
hyperbolas
Find an equation for the hyperbola whose graph has vertices (+/5,0) and foci (+/7,0) 
math
find the equation of the hyperbola with vertices (4,0), (4,0) and focus (7,0)