Answer :
To determine the correct equation of the hyperbola [tex]\(\frac{(y-2)^2}{4} - \frac{(x-2)^2}{9} = 1\)[/tex] in its general form, we need to manipulate and simplify this given equation. Let's go through this step-by-step:
1. Starting with the given equation:
[tex]\[ \frac{(y-2)^2}{4} - \frac{(x-2)^2}{9} = 1 \][/tex]
2. Clear the fractions by multiplying every term by the least common multiple of the denominators, which is 36:
[tex]\[ 36 \left( \frac{(y-2)^2}{4} - \frac{(x-2)^2}{9} \right) = 36 \cdot 1 \][/tex]
[tex]\[ 9(y-2)^2 - 4(x-2)^2 = 36 \][/tex]
3. Expand the terms:
[tex]\[ 9(y^2 - 4y + 4) - 4(x^2 - 4x + 4) = 36 \][/tex]
4. Distribute the constants 9 and -4 inside the parentheses:
[tex]\[ 9y^2 - 36y + 36 - 4x^2 + 16x - 16 = 36 \][/tex]
5. Move all the terms to one side of the equation to set it equal to zero:
[tex]\[ 9y^2 - 4x^2 - 36y + 16x + 36 - 16 - 36 = 0 \][/tex]
6. Combine the constant terms:
[tex]\[ 9y^2 - 4x^2 - 36y + 16x - 16 = 0 \][/tex]
So, the general form of the given hyperbola is:
[tex]\[ 9y^2 - 4x^2 - 36y + 16x - 16 = 0 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{9y^2 - 4x^2 - 36y + 16x - 16 = 0} \][/tex]
This matches option D. Therefore, the correct answer is:
D. [tex]\(9 y^2-4 x^2-36 y+16 x-16=0\)[/tex]
1. Starting with the given equation:
[tex]\[ \frac{(y-2)^2}{4} - \frac{(x-2)^2}{9} = 1 \][/tex]
2. Clear the fractions by multiplying every term by the least common multiple of the denominators, which is 36:
[tex]\[ 36 \left( \frac{(y-2)^2}{4} - \frac{(x-2)^2}{9} \right) = 36 \cdot 1 \][/tex]
[tex]\[ 9(y-2)^2 - 4(x-2)^2 = 36 \][/tex]
3. Expand the terms:
[tex]\[ 9(y^2 - 4y + 4) - 4(x^2 - 4x + 4) = 36 \][/tex]
4. Distribute the constants 9 and -4 inside the parentheses:
[tex]\[ 9y^2 - 36y + 36 - 4x^2 + 16x - 16 = 36 \][/tex]
5. Move all the terms to one side of the equation to set it equal to zero:
[tex]\[ 9y^2 - 4x^2 - 36y + 16x + 36 - 16 - 36 = 0 \][/tex]
6. Combine the constant terms:
[tex]\[ 9y^2 - 4x^2 - 36y + 16x - 16 = 0 \][/tex]
So, the general form of the given hyperbola is:
[tex]\[ 9y^2 - 4x^2 - 36y + 16x - 16 = 0 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{9y^2 - 4x^2 - 36y + 16x - 16 = 0} \][/tex]
This matches option D. Therefore, the correct answer is:
D. [tex]\(9 y^2-4 x^2-36 y+16 x-16=0\)[/tex]
