Let’s analyze and compare the two hyperbolas given by the equations:
1. [tex]\(\frac{x^2}{3^2} - \frac{y^2}{4^2} = 1\)[/tex]
2. [tex]\(\frac{y^2}{3^2} - \frac{x^2}{4^2} = 1\)[/tex]
Step 1: Identify the type and properties of each equation
For [tex]\(\frac{x^2}{3^2} - \frac{y^2}{4^2} = 1\)[/tex]:
- This is of the form [tex]\(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)[/tex] which represents a horizontal hyperbola.
- Vertices are located at [tex]\((\pm3, 0)\)[/tex] on the x-axis.
- Foci can be found using [tex]\(c = \sqrt{a^2 + b^2}\)[/tex]. Here, [tex]\(a = 3\)[/tex] and [tex]\(b = 4\)[/tex], so [tex]\(c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\)[/tex].
- Transverse axis length is [tex]\(2a = 2 \cdot 3 = 6\)[/tex].
For [tex]\(\frac{y^2}{3^2} - \frac{x^2}{4^2} = 1\)[/tex]:
- This is of the form [tex]\(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\)[/tex] which represents a vertical hyperbola.
- Vertices are located at [tex]\((0, \pm3)\)[/tex] on the y-axis.
- Foci can be found using [tex]\(c = \sqrt{a^2 + b^2}\)[/tex]. Here, [tex]\(a = 3\)[/tex] and [tex]\(b = 4\)[/tex], so [tex]\(c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\)[/tex].
- Transverse axis length is [tex]\(2a = 2 \cdot 3 = 6\)[/tex].
Step 2: Compare the properties
- Foci: Both hyperbolas have the same foci, which are located at a distance of 5 units from the center along the transverse axis.
- Transverse axis: Both hyperbolas have the same length of the transverse axis, which is 6 units.
- Directrices: The horizontal hyperbola [tex]\(\frac{x^2}{3^2} - \frac{y^2}{4^2} = 1\)[/tex] has vertical directrices, while the vertical hyperbola [tex]\(\frac{y^2}{3^2} - \frac{x^2}{4^2} = 1\)[/tex] has horizontal directrices.
- Vertices: For [tex]\(\frac{x^2}{3^2} - \frac{y^2}{4^2} = 1\)[/tex], the vertices are on the x-axis ([tex]\(\pm3, 0\)[/tex]). For [tex]\(\frac{y^2}{3^2} - \frac{x^2}{4^2} = 1\)[/tex], the vertices are on the y-axis ([tex]\(0, \pm3\)[/tex]).
Conclusion
From the above details, the correct statements are:
1. The foci of both graphs are the same points.
2. The lengths of both transverse axes are the same.
Therefore, the correct descriptions for these hyperbolas are the first and second options:
- The foci of both graphs are the same points.
- The lengths of both transverse axes are the same.
