1. Since the mirror has a vertex 5 inches from the center of the hyperbola and a focus 1 inch in front of the surface of the mirror, the equation of the hyperbola can be written as:
\( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)
Where the distance from the vertex to the center is a = 5 inches and the distance from the center to the focus is c = 1 inch. From the properties of a hyperbola, we know that \( c^2 = a^2 + b^2 \).
Substitute the values of a and c into the equation:
\( 1^2 = 5^2 + b^2 \)
\( 1 = 25 + b^2 \)
\( b^2 = -24 \)
Thus, the equation of the hyperbola is:
\( \frac{x^2}{25} - \frac{y^2}{24} = 1 \)
2. The equation given for the elliptical track is:
\( \frac{x^2}{16} + \frac{y^2}{25} = 1 \)
This is the standard form of an ellipse centered at (0, 0). The major axis of an ellipse is the longest diameter of the ellipse, which passes through the center. In this case, the major axis is along the x-axis, so it is the distance from (-4, 0) to (4, 0), which is equal to 8 yards.
To find this distance, we know that the equation of the ellipse can be written as:
\( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)
The major axis is equal to 2a. Therefore, in this case, the major axis length is 2(4) = 8 yards.
3. The equation of a parabola is given by \( y = x^2 + 4x - 3 \).
To identify the vertex of the parabola, we first rewrite the equation in vertex form:
\( y = (x + 2)^2 - 7 \)
From this form, we can see that the vertex is at (-2, -7).
To find the focus of the parabola, we know that the focus is located at (h, k + 1/(4a)), where (h, k) is the vertex and a is the coefficient of the x^2 term. In this case, a = 1. Therefore, the focus is at (-2, -7 + 1/(4*1)) = (-2, -6.75).
The directrix of the parabola is a horizontal line passing through (h, k - 1/(4a)). In this case, the directrix is at y = -7 - 1/(4*1) = -7.25.
1. A hyperbolic mirror can be used to take panoramic photos, if the camera is pointed toward the mirror with the lens at one focus of the hyperbola. Write the equation of the hyperbola that can be used to model a mirror that has a vertex 5 inches from the center of the hyperbola and a focus 1 inch in front of the surface of the mirror. Assume the mirror has a horizontal transverse axis and the hyperbola is centered at (0, 0).
2. Skip designs tracks for amusement park rides. For a new design, the track will be elliptical. If the ellipse is placed on a large coordinate grid with its center at (0, 0), the equation models the path of the track. The units are given in yards. How long is the major axis of the track? Explain how you found the distance.
3. The equation of a parabola is . Identify the vertex, focus, and directrix of the parabola.
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