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parabola reflective property question?
How do I prove that tan B = 2p / y0 ?
2 Answers
- PopeLv 73 years ago
Line L includes point P(x₀, y₀).
Projecting that point onto the axis, its image is Q(x₀, 0).
Reflecting Q through the vertex, its image is R(-x₀, 0).
Line L includes points P and R.
slope(line L)
= slope(PR)
= y₀/(2x₀)
Since point P is on the parabola, its coordinates must satisfy the equation.
y₀² = 4px₀
2x₀ = y₀²/(2p)
slope(line L)
= y₀/(2x₀)
= y₀/[y₀²/(2p)]
= 2p/y₀
tan(β) = slope(line L) = 2p/y₀
- ?Lv 73 years ago
y^2 = 4px
2ydy/dx = 4p
In general, dy/dx = 2p/y
At the particular point P, y = yo
tan(B) = dy/dx = 2p/yo
