SOLUTION: The reflection property of parabolas. Consider the parabola whose focus is F = (1,4) and whose directrix is the line x = −3. (a) Sketch the parabola, and make calculations that

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Question 1157992: The reflection property of parabolas. Consider the parabola whose focus is F = (1,4) and whose directrix is the line x = −3.
(a) Sketch the parabola, and make calculations that confirm that P = (7, 12) is on it.
(b) Find the slope of the line μ through P that is tangent to the parabola.
(c) Calculate the size of the angle that μ makes with the line y = 12.
(d) Calculate the size of the angle that μ makes with segment F P .
Answer by KMST(5377) About Me  (Show Source):
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(a) Sketch the parabola, and make calculations that confirm that P = (7, 12) is on it.

A parabola is the locus of the points that are at the same distance from the directrix and the focus.
The distance between directrix x=-3 and P%287%2C12%29 is
7-%28-3%29=7%2B3=10
The distance between focus F%281%2C4%29 and P%287%2C12%29 is

P%287%2C12%29 is at the same distance from the directrix and the focus,
so it is on the parabola.
We could prove P is on the parabola from the equation of the parabola.
Knowing that the equation of a parabola with its vertex at the origin and focal distance f is
x=%281%2F4f%29y%5E2 for parabolas with the x-axis as an axis of symmetry,
we can translate that equation for a parabola with the vertex at (-1,4), halfway between focus and directrix.
Doing that, we found that our parabola has f=1-%28-1%29=2 and the equation for our parabola would be
x-%28-1%29=%281%2F%284%2A2%29%29%28y-4%29%5E2-->x%2B1=%281%2F8%29%28y-4%29%5E2
For y=12 we get x%2B1=%281%2F8%29%2812-4%29%5E2=%281%2F8%29%2A8%5E2=8-->x=8-1=7

(b) Find the slope of the line μ through P that is tangent to the parabola.
We can estimate the slope of the tangent line from the graph or calculate it from the derivative of the function.
From x%2B1=%281%2F8%29%28y-4%29%5E2-->x=%281%2F8%29%28y-4%29%5E2-1 we can find the derivative
dx%2Fdy=%281%2F8%29%2A2%28y-4%29 .
For P%287%2C12%29 , that derivative is dx%2Fdy=%281%2F8%29%2A2%2812-4%29=%281%2F8%29%2A2%2A8=2
For the function graphed in red above, dy%2Fdx=highlight%281%2F2%29 is the derivative and slope of the tangent at point P.

(c) Calculate the size of the angle that μ makes with the line y = 12.
(d) Calculate the size of the angle that μ makes with segment F P .

The line y=12 slope is zero, it is parallel to the x-axis.
Line blue%28mu%29 with slope 1%2F2 makes an angle green%28alpha%29 with the x-axis and with the liney=12 such that
tan%28green%28alpha%29%29=1%2F2 --> green%28alpha%29=highlight%2826.565%5Eo%29
Segment FP, connecting F%281%2C4%29} and P%287%2C12%29 , and line FP, have a slope of
%2812-4%29%2F%287-1%29=8%2F6=4%2F3 . A line, or segment with such a slope would make an angle green%28beta%29 with the x-axis and line y=12 such that
tan%28green%28beta%29%29=4%2F3 --> green%28beta%29=53.130%5Eo
The angle that μ makes with segment FP is
green%28beta%29-green%28alpha%29=53.130%5Eo-26.565%5Eo=highlight%2826.565%5Eo%29