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To solve the problem of identifying the center, vertices, foci, and asymptotes of the hyperbola given by the equation [tex]\( x^2 - y^2 = 10 \)[/tex], follow these steps:
### Step 1: Rewrite the Equation in Standard Form
The given equation is:
[tex]\[ x^2 - y^2 = 10 \][/tex]
First, recognize that the given equation is already in the form of a hyperbola. To make it clearer, we can rewrite it as:
[tex]\[ \frac{x^2}{10} - \frac{y^2}{10} = 1 \][/tex]
This matches the standard form of a hyperbola:
[tex]\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \][/tex]
where [tex]\( a^2 = 10 \)[/tex] and [tex]\( b^2 = 10 \)[/tex].
### Step 2: Determine the Center
The center of the hyperbola is given by the coordinates ([tex]\(h, k\)[/tex]). Since the equation [tex]\( x^2 - y^2 = 10 \)[/tex] does not have any shifting terms (no [tex]\( (x - h)^2 \)[/tex] or [tex]\( (y - k)^2 \)[/tex]), the center is at:
[tex]\[ (h, k) = (0, 0) \][/tex]
### Step 3: Identify the Vertices
The vertices of the hyperbola lie along the transverse axis (the [tex]\(x\)[/tex]-axis in this case) and are located at ([tex]\(\pm a, 0\)[/tex]).
Here, [tex]\( a^2 = 10 \)[/tex], thus [tex]\( a = \sqrt{10} \)[/tex]. Therefore, the vertices are:
[tex]\[ \left( -\sqrt{10}, 0 \right) \quad \text{and} \quad \left( \sqrt{10}, 0 \right) \][/tex]
Numerically, this is approximately:
[tex]\[ (-3.16227766016838, 0) \quad \text{and} \quad (3.16227766016838, 0) \][/tex]
### Step 4: Identify the Foci
The foci are determined by the relationship [tex]\( c^2 = a^2 + b^2 \)[/tex]. Here, both [tex]\( a^2 \)[/tex] and [tex]\( b^2 \)[/tex] are equal to 10.
[tex]\[ c = \sqrt{a^2 + b^2} = \sqrt{10 + 10} = \sqrt{20} \approx 4.47213595499958 \][/tex]
Thus, the foci are located at ([tex]\(\pm c, 0\)[/tex]):
[tex]\[ (-4.47213595499958, 0) \quad \text{and} \quad (4.47213595499958, 0) \][/tex]
### Step 5: Find the Asymptotes
The asymptotes of the hyperbola are given by the equations:
[tex]\[ y = \pm \frac{b}{a} x \][/tex]
Given [tex]\( b = \sqrt{10} \)[/tex] and [tex]\( a = \sqrt{10} \)[/tex], the asymptotes simplify to:
[tex]\[ y = \pm x \][/tex]
Written as explicit equations:
[tex]\[ y = x \quad \text{and} \quad y = -x \][/tex]
### Summary
- Center: [tex]\((0, 0)\)[/tex]
- Vertices: [tex]\((-3.16227766016838, 0)\)[/tex] and [tex]\((3.16227766016838, 0)\)[/tex]
- Foci: [tex]\((-4.47213595499958, 0)\)[/tex] and [tex]\((4.47213595499958, 0)\)[/tex]
- Asymptotes: [tex]\( y = x \)[/tex] and [tex]\( y = -x \)[/tex]
This concludes the detailed solution for the given hyperbola.
### Step 1: Rewrite the Equation in Standard Form
The given equation is:
[tex]\[ x^2 - y^2 = 10 \][/tex]
First, recognize that the given equation is already in the form of a hyperbola. To make it clearer, we can rewrite it as:
[tex]\[ \frac{x^2}{10} - \frac{y^2}{10} = 1 \][/tex]
This matches the standard form of a hyperbola:
[tex]\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \][/tex]
where [tex]\( a^2 = 10 \)[/tex] and [tex]\( b^2 = 10 \)[/tex].
### Step 2: Determine the Center
The center of the hyperbola is given by the coordinates ([tex]\(h, k\)[/tex]). Since the equation [tex]\( x^2 - y^2 = 10 \)[/tex] does not have any shifting terms (no [tex]\( (x - h)^2 \)[/tex] or [tex]\( (y - k)^2 \)[/tex]), the center is at:
[tex]\[ (h, k) = (0, 0) \][/tex]
### Step 3: Identify the Vertices
The vertices of the hyperbola lie along the transverse axis (the [tex]\(x\)[/tex]-axis in this case) and are located at ([tex]\(\pm a, 0\)[/tex]).
Here, [tex]\( a^2 = 10 \)[/tex], thus [tex]\( a = \sqrt{10} \)[/tex]. Therefore, the vertices are:
[tex]\[ \left( -\sqrt{10}, 0 \right) \quad \text{and} \quad \left( \sqrt{10}, 0 \right) \][/tex]
Numerically, this is approximately:
[tex]\[ (-3.16227766016838, 0) \quad \text{and} \quad (3.16227766016838, 0) \][/tex]
### Step 4: Identify the Foci
The foci are determined by the relationship [tex]\( c^2 = a^2 + b^2 \)[/tex]. Here, both [tex]\( a^2 \)[/tex] and [tex]\( b^2 \)[/tex] are equal to 10.
[tex]\[ c = \sqrt{a^2 + b^2} = \sqrt{10 + 10} = \sqrt{20} \approx 4.47213595499958 \][/tex]
Thus, the foci are located at ([tex]\(\pm c, 0\)[/tex]):
[tex]\[ (-4.47213595499958, 0) \quad \text{and} \quad (4.47213595499958, 0) \][/tex]
### Step 5: Find the Asymptotes
The asymptotes of the hyperbola are given by the equations:
[tex]\[ y = \pm \frac{b}{a} x \][/tex]
Given [tex]\( b = \sqrt{10} \)[/tex] and [tex]\( a = \sqrt{10} \)[/tex], the asymptotes simplify to:
[tex]\[ y = \pm x \][/tex]
Written as explicit equations:
[tex]\[ y = x \quad \text{and} \quad y = -x \][/tex]
### Summary
- Center: [tex]\((0, 0)\)[/tex]
- Vertices: [tex]\((-3.16227766016838, 0)\)[/tex] and [tex]\((3.16227766016838, 0)\)[/tex]
- Foci: [tex]\((-4.47213595499958, 0)\)[/tex] and [tex]\((4.47213595499958, 0)\)[/tex]
- Asymptotes: [tex]\( y = x \)[/tex] and [tex]\( y = -x \)[/tex]
This concludes the detailed solution for the given hyperbola.
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