Certainly! Given the hyperbola equation:
[tex]\[ \frac{y^2}{49} - \frac{x^2}{36} = 1 \][/tex]
We recognize this as the standard form of a hyperbola with the equation:
[tex]\[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \][/tex]
The parameters we need to identify are [tex]\( a^2 \)[/tex] and [tex]\( b^2 \)[/tex].
### Step-by-Step Solution:
1. Identify [tex]\( a^2 \)[/tex]:
From the equation [tex]\( \frac{y^2}{49} - \frac{x^2}{36} = 1 \)[/tex], we can see that:
[tex]\[ \frac{y^2}{49} = \frac{y^2}{a^2} \][/tex]
Therefore, [tex]\( a^2 = 49 \)[/tex].
2. Identify [tex]\( b^2 \)[/tex]:
Similarly, we see that:
[tex]\[ \frac{x^2}{36} = \frac{x^2}{b^2} \][/tex]
Therefore, [tex]\( b^2 = 36 \)[/tex].
3. Calculate [tex]\( a \)[/tex]:
To find [tex]\( a \)[/tex], we take the square root of [tex]\( a^2 \)[/tex]:
[tex]\[ a = \sqrt{49} = 7 \][/tex]
4. Calculate [tex]\( b \)[/tex]:
To find [tex]\( b \)[/tex], we take the square root of [tex]\( b^2 \)[/tex]:
[tex]\[ b = \sqrt{36} = 6 \][/tex]
So, one way to fill the two parts in the hyperbola equation could be:
[tex]\[ \square = 49, \][/tex]
and:
[tex]\[ \square = 36. \][/tex]
Also, you may want:
[tex]\[ \square = 7.0,\][/tex]
and:
[tex]\[ \square = 6.0.]
Thus, the correct values for:
\[ \frac{y^2}{\square} - \frac{x^2}{\square} = 1 \][/tex]
And:
[tex]\[
\square = a,
\][/tex]
[tex]\[
\square = b.
]
Are:
\[ 49 \quad \text{for} \quad y^2/\square\][/tex]
And:
[tex]\[ 36 \quad \text{for} \quad x^2/\square.\][/tex]
And:
[tex]\( 7 \quad \text{for\)[/tex] \quad\]
And:
[tex]\[ 6 \quad \text{for\][/tex] \quad].
So,
\]:
\:
And:
\
