Determine Bulb Position for Parabolic Reflector of Headlight

[runicle]

The cross section of a parabolic reflector of a headlight is shown in the diagram. The distance from B to C is 32 cm and the distance from A to D is 8 cm. Determine where the builb should be located if it is positioned at the focus.

. ...|...B
. ...|
. ...|
. ...|
----|---------D
. ...|A
. ...|
. ...|
. ...|...C

Remember its a parabola connect the dots.


This is what i have done
x^2/a^2 - y^2/b^2 = 1
then i substituted a as 0 and b as 16 and it got me nowhere
I'm guessing it has to be located at 0 but i don't know how to prove it all i know are these equations...
x^2/a^2 - y^2/b^2 = 1, a^2 +b^2 = c^2 and that's all
(P.S. Latex is hard to learn)

 

[Andrew Mason]

This is what i have done
x^2/a^2 - y^2/b^2 = 1
then i substituted a as 0 and b as 16 and it got me nowhere
I'm guessing it has to be located at 0 but i don't know how to prove it all i know are these equations...
x^2/a^2 - y^2/b^2 = 1, a^2 +b^2 = c^2 and that's all
(P.S. Latex is hard to learn)

The equation for a parabola in this orientation is:

[tex]y^2 = 4px[/tex] where A = (0,0) is the origin and p is the distance from the origin to the focus. The points B = (16, 16), C = (16, -16) are on the parabola. That will enable you to solve for p.

AM

 

[kyle8921]

I would approach this problem by first understanding the basic principles of parabolic reflectors and their use in headlights. A parabolic reflector is designed such that all light rays parallel to the axis of the parabola will reflect and converge at a single point, known as the focus. This property makes it ideal for use in headlights, where the light from the bulb needs to be focused in a specific direction.

In this case, we are given the distance from B to C as 32 cm and the distance from A to D as 8 cm. We can use these values to determine the focal length of the parabolic reflector, which is equal to half the distance from B to C. Therefore, the focal length in this case is 16 cm.

Now, we need to determine where the bulb should be located in relation to the parabolic reflector. Since all light rays parallel to the axis of the parabola will reflect and converge at the focus, the bulb should be positioned at the focus. In this case, the focus is located at the origin (0,0) since the parabola is symmetrical.

To prove this mathematically, we can use the equation of a parabola, which is y^2 = 4px, where p is the focal length. Substituting p as 16 cm, we get y^2 = 64x. This means that for any value of x, the corresponding value of y will be the square root of 64x. This confirms that the bulb should be located at the origin, as all light rays parallel to the axis will converge at this point.

In conclusion, the bulb should be located at the focus, which is at the origin (0,0), for optimal performance of the parabolic reflector in a headlight.