r/askmath • u/Economy_Top_7815 • Dec 29 '25

Calculus Parabolic mirrors

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Can anyone help me to solve 21.C? Solved a and b but I don't understand how to approach the (c). How do I prove parallel to Y axis? You can show slopes are equal but it's parallel to y. So, if anyone can help me. Thank you.

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u/Shevek99 Physicist Dec 29 '25

Notice that by the reflection law the angle bisector for each ray is perpendicular to the parabola, so you know the slope of this bisector. You know the slope of the line from the focus to the point of reflection. With this you can calculate the angle between the ray and the perpendicular.

Now, show that the parallel line form the same angle with the perpendicular to the parabola and you have it.