To solve the problem of finding the new center of the hyperbola after it has been shifted, follow these steps:
1. Identify the original center:
The given equation of the hyperbola is [tex]\(\frac{(x-5)^2}{7} - \frac{(y+3)^2}{9} = 1\)[/tex]. This equation is in standard form for a hyperbola centered at [tex]\((h, k)\)[/tex].
Comparing it with the standard form [tex]\(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\)[/tex], we identify:
- [tex]\(h = 5\)[/tex]
- [tex]\(k = -3\)[/tex]
Therefore, the original center of the hyperbola is [tex]\((5, -3)\)[/tex].
2. Apply the horizontal shift:
The hyperbola is shifted 4 units to the right. To find the new [tex]\(x\)[/tex]-coordinate of the center:
- Original [tex]\(x\)[/tex]-coordinate: 5
- Shift to the right: +4
Therefore, the new [tex]\(x\)[/tex]-coordinate is [tex]\(5 + 4 = 9\)[/tex].
3. Apply the vertical shift:
The hyperbola is shifted 3 units upward. To find the new [tex]\(y\)[/tex]-coordinate of the center:
- Original [tex]\(y\)[/tex]-coordinate: -3
- Shift upward: +3
Therefore, the new [tex]\(y\)[/tex]-coordinate is [tex]\(-3 + 3 = 0\)[/tex].
Combining these two results, the new center of the hyperbola is [tex]\((9, 0)\)[/tex].
Hence, the new center of the hyperbola is [tex]\(( \ 9 \ , \ 0 \ )\)[/tex].
