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The vertices of a hyperbola are (2, 0), (-2, 0) and the foci are (3, 0), (-3, 0). The equation of the hyperbola is
- a)x2/5 - y2/4 = 1
- b)x2/4 - y2/5 = 1
- c)x2/5 - y2/2 = 1
- d)x2/2 - y2/5 = 1
Correct answer is option 'B'. Can you explain this answer?
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The vertices of a hyperbola are (2, 0), (-2, 0) and the foci are (3, 0...
Given Data:
- Vertices: (2, 0), (-2, 0)
- Foci: (3, 0), (-3, 0)
Formula for Hyperbola:
The standard form of the equation of a hyperbola centered at the origin is:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
Finding the values of a and b:
- The distance between the vertices gives the value of 2a, which is 4. So, a = 2.
- The distance between the foci gives the value of 2ae, where e is the eccentricity of the hyperbola. Since e = 1, the distance between the foci is 6. Therefore, 2ae = 6, and we already know a = 2. Thus, e = 1.5.
- The value of b can be calculated using the formula b = a * sqrt(e^2 - 1). Substituting the values, we get b = 2 * sqrt(1.5^2 - 1) = 2 * sqrt(2.25 - 1) = 2 * sqrt(1.25) = 2 * 1.118 = 2.236.
Equation of the Hyperbola:
Substitute the values of a and b into the standard form equation:
\[ \frac{x^2}{4} - \frac{y^2}{5} = 1 \]
Therefore, the correct equation of the hyperbola is option 'b':
\[ \frac{x^2}{4} - \frac{y^2}{5} = 1 \]
- Vertices: (2, 0), (-2, 0)
- Foci: (3, 0), (-3, 0)
Formula for Hyperbola:
The standard form of the equation of a hyperbola centered at the origin is:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
Finding the values of a and b:
- The distance between the vertices gives the value of 2a, which is 4. So, a = 2.
- The distance between the foci gives the value of 2ae, where e is the eccentricity of the hyperbola. Since e = 1, the distance between the foci is 6. Therefore, 2ae = 6, and we already know a = 2. Thus, e = 1.5.
- The value of b can be calculated using the formula b = a * sqrt(e^2 - 1). Substituting the values, we get b = 2 * sqrt(1.5^2 - 1) = 2 * sqrt(2.25 - 1) = 2 * sqrt(1.25) = 2 * 1.118 = 2.236.
Equation of the Hyperbola:
Substitute the values of a and b into the standard form equation:
\[ \frac{x^2}{4} - \frac{y^2}{5} = 1 \]
Therefore, the correct equation of the hyperbola is option 'b':
\[ \frac{x^2}{4} - \frac{y^2}{5} = 1 \]
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The vertices of a hyperbola are (2, 0), (-2, 0) and the foci are (3, 0), (-3, 0). The equation of the hyperbola isa) x2/5 - y2/4 = 1 b) x2/4 - y2/5 = 1 c) x2/5 - y2/2 = 1 d) x2/2 - y2/5 = 1 Correct answer is option 'B'. Can you explain this answer?
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