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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).

You are watching: Find an equation in standard form for the hyperbola with vertices at (0, ±2) and foci at (0, ±11).

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

Given hyperbola

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).

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**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).

2*k* = 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

*c*2 = *a*2 + *b*2

(7)2 = (2)2 + *b*2

49 – 4 = *b*2

*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.

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Therefore, the standard form of hyperbola is **(y – 0)2/4 – (x – 0)2/45 = 1.**

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