Let's analyze and compare the given equations:
1. [tex]\(\frac{x^2}{6^2} - \frac{y^2}{8^2} = 1\)[/tex]
2. [tex]\(\frac{x^2}{8^2} - \frac{y^2}{6^2} = 1\)[/tex]
These equations represent hyperbolas. To compare the characteristics, we need to look at different aspects such as the foci, the lengths of the transverse axes, the directions of the directrices, and the positions of the vertices.
### Equation 1: [tex]\(\frac{x^2}{6^2} - \frac{y^2}{8^2} = 1\)[/tex]
1. Vertices:
- The hyperbola opens horizontally because the [tex]\(x^2\)[/tex] term is positive.
- The vertices are at [tex]\(( \pm 6, 0 )\)[/tex] because [tex]\(a^2 = 6^2 = 36\)[/tex].
2. Transverse Axis:
- The length is [tex]\(2a = 2 \times 6 = 12\)[/tex].
3. Directrices:
- Horizontal hyperbolas have vertical directrices.
### Equation 2: [tex]\(\frac{x^2}{8^2} - \frac{y^2}{6^2} = 1\)[/tex]
1. Vertices:
- The hyperbola also opens horizontally because the [tex]\(x^2\)[/tex] term is positive.
- The vertices are at [tex]\(( \pm 8, 0 )\)[/tex] because [tex]\(a^2 = 8^2 = 64\)[/tex].
2. Transverse Axis:
- The length is [tex]\(2a = 2 \times 8 = 16\)[/tex].
3. Directrices:
- Horizontal hyperbolas have vertical directrices.
### Comparing the Statements:
1. The foci of both graphs are the same points.
- This is false because the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are different in both equations, leading to different foci.
2. The lengths of both transverse axes are the same.
- This is false. The length of the transverse axis for the first equation is 12, while for the second equation it is 16.
3. The directrices of [tex]\(\frac{x^2}{6^2} - \frac{y^2}{8^2} = 1\)[/tex] are horizontal while the directrices of [tex]\(\frac{x^2}{8^2} - \frac{y^2}{6^2} = 1\)[/tex] are vertical.
- This is true. The directrices in a horizontal hyperbola are vertical lines, hence the directrices for the first equation (horizontal hyperbola) are horizontal, and for the second equation (horizontal hyperbola), they are vertical due to their positions as determined by the axes’ lengths and orientations.
4. The vertices of [tex]\(\frac{x^2}{6^2} - \frac{y^2}{8^2} = 1\)[/tex] are on the [tex]\(y\)[/tex]-axis while the vertices of [tex]\(\frac{x^2}{8^2} - \frac{y^2}{6^2} = 1\)[/tex] are on the [tex]\(x\)[/tex]-axis.
- This is false because both hyperbolas have their vertices on the [tex]\(x\)[/tex]-axis since they open horizontally.
Therefore, the true statement is:
The directrices of [tex]\(\frac{x^2}{6^2} - \frac{y^2}{8^2} = 1\)[/tex] are horizontal while the directrices of [tex]\(\frac{x^2}{8^2} - \frac{y^2}{6^2} = 1\)[/tex] are vertical.
That is the correct statement based on the given equations.
