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Find an equation in standard form for the hyperbola with 0 votes
vertices at (0, ±2) and foci at (0, ±7).
hyperbolaaskedJul 11, 2013in PRECALCULUSby homeworkhelpMentor
Please log in or register to add a comment. Please log in or register to answer this question. 2 Answers 0 votes
General equation of a hyperbola
We have foci and vertices are on the y-axis, which means that we needs the formula for a up and down hyperbola.
This means that the center (h, k) must be a the origin, or (0, 0). So, let”s label that…
h = 0k = 0
We know that “a” is the distance from your vertex and “c” is the distance from your foci
a = 2 and c = 7
We have a formula that
Now fill that
h = 0
k = 0
Therefore the required equation of hyperbola is
answeredJul 11, 2013by jouisApprentice Please log in or register to add a comment. 0 votes
The vertices of the hyperbola are (0, 2) and (0, – 2) and its foci are (0, 7) and (0, – 7).
Since the x – coordinate isconstantin theverticesandfoci, this is a vertical hyperbola.
The standard form of vertical hyperbola (y – k)2/a2 – (x – h)2/b2 = 1.
Where, “b ” is the number in the denominator of the positive term, If the x – term is negative, then the hyperbola is vertical.a = semi – transverse axis , b = semi – conjugate axis .Center: (h, k )Vertices: (h , k + a ) and (h, k – a).Foci: (h , k +c) and (h, k – c).
So, the x coodinate of the center of hyperbola is 0.
vertices : (0, 2) and (0,- 2)
k + a = 2 —-> (1)
k – a = – 2 —> (2)
Add the equations (1) & (2).
2k = 0
⇒ k = 0
So,the y coordinate of center is 0.
Substitute the k value in (1),
0 + a = 2
⇒ a = 2.
foci : (0, 7) and (0, – 7)
k + c = 7
0 + c = 7
c = 7
c2 = a2 + b2
(7)2 = (2)2 + b2
49 – 4 = b2
b = √45
Substitute the (h , k), a, and b in standard form of hyperbola equation .
(y – 0)2/22 – (x – 0)2/(√45)2 = 1
(y – 0)2/4 – (x – 0)2/45 = 1.
Therefore, the standard form of hyperbola is (y – 0)2/4 – (x – 0)2/45 = 1.
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