The equation of the conjugate axis of the hyperbola
`((y-2)^(2))/(9)-((x+3)^(2))/(16)=1` is

Options

A. y=2
B. y=6
C. y=8
D. y=3

Answer

To find the equation of the conjugate axis of the given hyperbola, we will follow these steps:

### Step 1: Identify the standard form of the hyperbola
The given equation of the hyperbola is:
\[ \frac{(y-2)^2}{9} - \frac{(x+3)^2}{16} = 1 \]
This is in the standard form of a hyperbola which is:
\[ \frac{(y-k)^2}{b^2} - \frac{(x-h)^2}{a^2} = 1 \]
where \((h, k)\) is the center of the hyperbola, \(b^2 = 9\) and \(a^2 = 16\).

### Step 2: Determine the center of the hyperbola
From the standard form, we can identify:
- \(k = 2\)
- \(h = -3\)

Thus, the center of the hyperbola is at the point \((-3, 2)\).

### Step 3: Write the equation of the conjugate axis
For a hyperbola of the form:
\[ \frac{(y-k)^2}{b^2} - \frac{(x-h)^2}{a^2} = 1 \]
the equation of the conjugate axis is given by:
\[ y - k = 0 \]
Substituting \(k = 2\) into this equation gives:
\[ y - 2 = 0 \]
or
\[ y = 2 \]

### Final Answer
The equation of the conjugate axis of the hyperbola is:
\[ y = 2 \]

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