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 4 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).
Thanks for this! Remember, it's not cheating if you're not caught.
Well, what's the process of it? I really need to learn how to do it. Even though I'm kinda lazy for maths, I want to learn.
well, you are told that
a = 4
c = 5
so, b = 3
The horizontal axis means
x^2/16 - y^2/9 = 1
So, given a, b, and c, we can plug it in.
The equation is (x+h)^2/a^2 - (y+k)^2/b^2 =1
So, when you have a center at (0,0), h=0 and k=0
a=4 b=3 c=5
You find b by using the Pythagorean Theorem.
c^2=a^2+b^2
25=16+b^2
Subtract 16 from both sides.
9=b^2
The square root of 9 is 3 so b=3
So, with that you have (x+0)^2/4^2 - (y+0)^2/3^2 =1
When you simplify that you get x^2/16 - y^2/9 =1
That is your final answer. I hope this helps for anyone looking for the full way to do this. Have a good day! :)
2. You're given the formula, and if you graph it, you can see it is a vertical ellipse. The base formula is (x - h)²/a² + (y - k)²/b². The major axis is 2a. The minor axis is 2b. Because a² = 2500, we have to use the square root on it and we get 50. Multiply that by 2, and the major axis is 100 units.
3. The equation of a vertical parabola is y = a (x - h)² +k
Our given equation is 12y = (x - 1)² -48. We need to isolate the 12, so we divide everything and get y = (1/12)(x - 1)² - 4.
The vertex is (h, k). If you look at the equation placement, h = 1 and k = -4. The vertex is (1, -4). The focus is (h, k + (1/(4a))). Put everything to get (1, -4 + (1/(4(1/12)))). Simplify it to (1, -1). The directrix is similar to the focus, so be careful! It's (h, k - (1/(4a))). Again, put everything together so, (1, -4 - (1/(4(1/12)))). Simplify to (1, -7).
Put simply;
Vertex: (1, -4)
Focus: (1, -1)
Directrix: (1, -7)
If you do just copy everything, please change some wording, or you could get caught!
Thank you RAT! I'll see if I can put the rest of the answers on ^^.
thanks Tea.
You're welcome! If you have any more questions or need further explanations, feel free to ask.
To find the equation of the hyperbola that can be used to model the mirror, we need to understand the properties of a hyperbola and use the given information.
The standard form equation of a hyperbola with a horizontal transverse axis, centered at the origin (0, 0), and a focus (c, 0) is:
(x^2 / a^2) - (y^2 / b^2) = 1
where a is the distance from the center to the vertex, and c is the distance from the center to the focus.
Given information:
- The vertex is 4 inches from the center of the hyperbola. This means a = 4 inches.
- The focus is 1 inch in front of the surface of the mirror. This means c = 1 inch.
Substituting these values into the equation, we get:
(x^2 / 4^2) - (y^2 / b^2) = 1
Simplifying further, we have:
(x^2 / 16) - (y^2 / b^2) = 1
Now, we need to find the value of b, which represents the distance from the center to the intersection of the hyperbola with the transverse axis. In this case, the transverse axis is the mirror's surface.
Since the focus is 1 inch in front of the surface of the mirror, and the vertex is 4 inches from the center, we can deduce that the distance from the vertex to the intersection point is (4 - 1) = 3 inches.
Therefore, b = 3 inches.
Substituting this value into the equation, the final equation of the hyperbola is:
(x^2 / 16) - (y^2 / 9) = 1
So, the equation of the hyperbola that can be used to model the mirror is (x^2 / 16) - (y^2 / 9) = 1.