For a hyperbolic mirror the two foci are 44 cm apart. The distance of the vertex from one focus is 6 cm and from the other focus is 38 cm. Position a coordinate system with the origin at the center of the hyperbola and with the
Foci on the y-axis. Find the equation of the hyperbola.
A) $\frac { y ^ { 2 } } { 256 } - \frac { x ^ { 2 } } { 228 } = 1$
B) $\frac { y ^ { 2 } } { 256 } - \frac { x ^ { 2 } } { 228 } = 5$
C) $\frac { x ^ { 2 } } { 256 } - \frac { y ^ { 2 } } { 228 } = 1$
D) $\frac { x ^ { 2 } } { 256 } - \frac { y ^ { 2 } } { 228 } = 5$

Question

For a hyperbolic mirror the two foci are 44 cm apart. The distance of the vertex from one focus is 6 cm and from the other focus is 38 cm. Position a coordinate system with the origin at the center of the hyperbola and with the
Foci on the y-axis. Find the equation of the hyperbola. 
A) $\frac { y ^ { 2 } } { 256 } - \frac { x ^ { 2 } } { 228 } = 1$
B) $\frac { y ^ { 2 } } { 256 } - \frac { x ^ { 2 } } { 228 } = 5$
C) $\frac { x ^ { 2 } } { 256 } - \frac { y ^ { 2 } } { 228 } = 1$
D) $\frac { x ^ { 2 } } { 256 } - \frac { y ^ { 2 } } { 228 } = 5$

Quizplus · Accepted Answer

The distance between the two foci of a hyperbola is $2c = 44$, so $c = 22$. The distance from the vertex to each focus is different, indicating the vertices are not equidistant from the foci, which is typical for a hyperbola. The distance from the center to a vertex (half the distance between vertices) is $a$, and can be found as the average of the distances from the center to each focus: $a = \frac{6 + 38}{2} = 22$. The relationship between $a$, $b$, and $c$ in a hyperbola is $c^2 = a^2 + b^2$. Here, $c = 22$, and $a = 22$, so $b^2 = c^2 - a^2 = 22^2 - 22^2 = 0$, which seems incorrect due to a misinterpretation of the distances given. The correct interpretation should consider the geometry of the hyperbola and the given distances correctly.Given the distances from the vertex to the foci, we should instead recognize that the total distance between the foci is the length of the major axis, and the distances given actually help us find $b^2$.Correcting the approach: The total distance between the foci is $2c = 44$, so $c = 22$. This is correct. However, the calculation of $a$ and $b$ needs to be aligned with the properties of a hyperbola. The distance from the center to a vertex is not directly given by the distances to the foci in the manner previously described. Instead, the correct approach to find $a$ and $b$ involves understanding that $a$ is half the distance between the vertices along the axis of the hyperbola, and $b$ is related to the shape's width.Given the setup, we actually know that $2a$ (the distance between the vertices) is not directly provided, but we can infer $a$ from the geometry described. The correct interpretation involves recognizing that the distances provided do not directly calculate $a$ as previously described. Instead, we should focus on the correct geometric interpretation of a hyperbola's elements.The mistake in the initial explanation was in misinterpreting the distances from the vertex to the foci as directly providing $a$ and $b$. Instead, these distances help define the hyperbola's orientation and size but do not directly translate to $a = 22$.Given the correct approach to defining a hyperbola, we know that $c = 22$, and we need to correctly identify $a$ and $b$ based on the hyperbola's standard equation and properties. The equation of a hyperbola on the y-axis is $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$, where $2a$ is the distance between the vertices along the y-axis, and $2b$ is the distance between the co-vertices along the x-axis.The correct calculation of $a$ and $b$ involves using the given distances and the hyperbola's properties correctly. However, the initial explanation incorrectly calculated $a$ and $b$, leading to confusion. The correct approach involves using the distances and the relationship $c^2 = a^2 + b^2$ correctly, considering the hyperbola's orientation and the given distances.Given the misinterpretation in the explanation, the focus should be on correctly applying the properties of hyperbolas to find $a$ and $b$, and then using these to write the correct equation of the hyperbola. The distances from the vertex to the foci provide information about the hyperbola's dimensions and orientation, but the initial explanation incorrectly applied these distances to calculate $a$ and $b$.The correct equation of the hyperbola, considering the orientation and properties of hyperbolas, should be determined by correctly applying the geometric properties and the given distances. The choice provided as correct (A) assumes a specific orientation and calculation of $a$ and $b$, but the explanation did not accurately describe the process to arrive at this equation based on the given information.

For a Hyperbolic Mirror the Two Foci Are 44 Cm y2256−x2228=1\frac { y ^ { 2 } } { 256 } - \frac { x ^ { 2 } } { 228 } = 1256y2​−228x2​=1

For a Hyperbolic Mirror the Two Foci Are 44 Cm $\frac { y ^ { 2 } } { 256 } - \frac { x ^ { 2 } } { 228 } = 1$