Answer :
To find the value that correctly fills in the blank in the given hyperbola equation \(\frac{x^2}{24^2} - \frac{y^2}{[\ldots]^2} = 1\), we will go through the necessary steps to determine the missing value step-by-step.
1. Identify the given values:
- The term \(\frac{x^2}{24^2}\) tells us that \(a^2 = 24^2\), so \(a = 24\).
- The directrix is given as \(x = \frac{576}{26}\).
2. Determine \(c\):
- For a hyperbola with the equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), and directrix \(x = \frac{a^2}{c}\), we can set up the equation for the directrix:
[tex]\[ x = \frac{a^2}{c} \][/tex]
- We know \(a^2 = 576\), so:
[tex]\[ \frac{576}{c} = \frac{576}{26} \][/tex]
- Solving for \(c\):
[tex]\[ c = 26 \][/tex]
3. Find \(b^2\):
- For hyperbolas, the relationship between \(a\), \(b\), and \(c\) is given by:
[tex]\[ c^2 = a^2 + b^2 \][/tex]
- Substitute the known values \(a = 24\) and \(c = 26\):
[tex]\[ 26^2 = 24^2 + b^2 \][/tex]
- Calculate \(26^2\) and \(24^2\):
[tex]\[ 676 = 576 + b^2 \][/tex]
- Solve for \(b^2\):
[tex]\[ b^2 = 676 - 576 = 100 \][/tex]
4. State the complete equation:
- Now that we have \(b^2 = 100\), we can fill in the blank in the original equation:
[tex]\[ \frac{x^2}{24^2} - \frac{y^2}{100} = 1 \][/tex]
Thus, the positive value that correctly fills in the blank in the equation [tex]\(\frac{x^2}{24^2} - \frac{y^2}{[\ldots]^2} = 1\)[/tex] is [tex]\(10\)[/tex] (since [tex]\(10^2 = 100\)[/tex]).
1. Identify the given values:
- The term \(\frac{x^2}{24^2}\) tells us that \(a^2 = 24^2\), so \(a = 24\).
- The directrix is given as \(x = \frac{576}{26}\).
2. Determine \(c\):
- For a hyperbola with the equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), and directrix \(x = \frac{a^2}{c}\), we can set up the equation for the directrix:
[tex]\[ x = \frac{a^2}{c} \][/tex]
- We know \(a^2 = 576\), so:
[tex]\[ \frac{576}{c} = \frac{576}{26} \][/tex]
- Solving for \(c\):
[tex]\[ c = 26 \][/tex]
3. Find \(b^2\):
- For hyperbolas, the relationship between \(a\), \(b\), and \(c\) is given by:
[tex]\[ c^2 = a^2 + b^2 \][/tex]
- Substitute the known values \(a = 24\) and \(c = 26\):
[tex]\[ 26^2 = 24^2 + b^2 \][/tex]
- Calculate \(26^2\) and \(24^2\):
[tex]\[ 676 = 576 + b^2 \][/tex]
- Solve for \(b^2\):
[tex]\[ b^2 = 676 - 576 = 100 \][/tex]
4. State the complete equation:
- Now that we have \(b^2 = 100\), we can fill in the blank in the original equation:
[tex]\[ \frac{x^2}{24^2} - \frac{y^2}{100} = 1 \][/tex]
Thus, the positive value that correctly fills in the blank in the equation [tex]\(\frac{x^2}{24^2} - \frac{y^2}{[\ldots]^2} = 1\)[/tex] is [tex]\(10\)[/tex] (since [tex]\(10^2 = 100\)[/tex]).
