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

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) (Show Source): You can put this solution on YOUR website!
(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 and is

The distance between focus and is

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 is
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 and the equation for our parabola would be
-->
For we get -->

(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 --> we can find the derivative
.
For , that derivative is
For the function graphed in red above, 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 slope is zero, it is parallel to the x-axis.
Line with slope makes an angle with the x-axis and with the line such that
-->
Segment FP, connecting } and , and line FP, have a slope of
. A line, or segment with such a slope would make an angle with the x-axis and line such that
-->
The angle that μ makes with segment FP is

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