well, you are told that
a = 4
c = 5
so, b = 3
The horizontal axis means
x^2/16 - y^2/9 = 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 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).
8 answers
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.
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! :)
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! :)
Thank you RAT! I'll see if I can put the rest of the answers on ^^.
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!
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!
thanks Tea.
You're welcome! If you have any more questions or need further explanations, feel free to ask.
1 answer