For what values of k does the graph of
`((x-2k)^(2))/(1)-((y-3k)^(2))/(3)=1` pass through the origin?

Options

A. only 0
B. only 1
C. `+-1`
D. `+-sqrt(5)`

Explanation

To determine the values of \( k \) for which the graph of the equation

\[ \frac{(x - 2k)^2}{1} - \frac{(y - 3k)^2}{3} = 1 \]

passes through the origin \((0, 0)\), we will substitute \( x = 0 \) and \( y = 0 \) into the equation and solve for \( k \).

### Step-by-Step Solution:

1. **Substitute the coordinates of the origin into the equation**:
\[ \frac{(0 - 2k)^2}{1} - \frac{(0 - 3k)^2}{3} = 1 \]

2. **Simplify the equation**:
\[ \frac{(2k)^2}{1} - \frac{(3k)^2}{3} = 1 \]
This simplifies to:
\[ \frac{4k^2}{1} - \frac{9k^2}{3} = 1 \]

3. **Combine the terms**:
The second term can be simplified:
\[ \frac{9k^2}{3} = 3k^2 \]
Therefore, the equation becomes:
\[ 4k^2 - 3k^2 = 1 \]

4. **Combine like terms**:
\[ k^2 = 1 \]

5. **Solve for \( k \)**:
Taking the square root of both sides gives:
\[ k = \pm 1 \]

### Final Answer:
The values of \( k \) for which the graph passes through the origin are \( k = 1 \) and \( k = -1 \).

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