Answer:
The equation of the hyperbola is
y
2
49
−
x
2
72
=
1
Explanation:
This is a hyperbola with a vertical transverse axis.
The general equation is
(
y
−
k
)
2
a
2
−
(
x
−
h
)
2
b
2
=
1
The center is
C
=
(
h
,
k
)
=
(
0
,
0
)
As the foci are
F
=
(
0
,
11
)
and
F
'
=
(
0
,
−
11
)
c
=
11
As the vertices are
A
=
(
0
,
7
)
and
A
'
=
(
0
,
−
7
)
a
=
7
And
b
2
=
c
2
−
a
2
=
11
2
−
7
2
=
121
−
49
=
72
The equation of the hyperbola is
y
2
49
−
x
2
72
=
1
graph{(y^2/49-x^2/72-1)=0 [-60.26, 56.84, -20.9, 37.6]}
An equation in standard form for a hyperbola with center (0, 0), vertex (0, 17), and focus (0, 19) is; y²/172 – x²/(6√2)² = 1
We are given that the hyperbola has;
Center (0, 0), Vertex (0, 17) and Focus (0, 19)
The general form of equation of the given hyperbola has a form of;
y²/a² - x²/b² = 1
Where;
±a is the y- coordinates of the vertices of the parabola (or y-intercepts).
b determines the asymptotes of the hyperbola in the equation y = ± (a/b)x.
From the vertex coordinates of (0.17), we have that; a = ± 17.
From the focus coordinates (0, 19), the y-coordinate of it is; c = 19 .
b can be found from Pythagorean theorem;
c² = a² + b²
Thus;
192 = 172 + b²
b² = 192 – 172
b² = 361 – 289
b = √72 = 6√2
The equation of the hyperbola is;
y²/172 – x²/(6√2)² = 1
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