If a light source is placed between the focus and the vertex of a parabolic mirror, the rays of light diverge after reflection, whereas if the light source is placed outside the focus, the rays...

If a light source is placed between the focus and the vertex of a parabolic mirror, the rays of light diverge after reflection, whereas if the light source is placed outside the focus, the rays converge after reflection. This is how car headlights can give a broad beam of light (high beam) from one filament and a narrow beam of light ( low beam) from a second filament. If the equation of a cross-sectional view of a parabolic mirror is  y^2=20x  , where x and y are in centimetres, what could be the distance between the vertex of the mirror and a filament if the light rays are to diverge after reflection from the mirror?

Asked on by lilian-0716

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embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

The equation for the cross-section of the paraboloid is given as `y^2=20x` . Since the light rays are to diverge, we need the light source to be placed between the focus and the vertex of the parabola.

The standard form for a parabola is `(y-k)^2=4p(x-h)` ; here h=k=0 so the parabola has vertex (0,0). p is the distance from the vertex to the focus (and to the directrix.)

For `y^2=20x` we have h=k=0 and p=5. The parabola opens to the right with vertex (0,0) and focus (5,0).

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The light source can be placed at a distance d, 0<d<5, from the vertex in order for the light rays to diverge.

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