SOLUTION: .Let A = (3,4) and B = (−3,4), which are both on the circle x2 +y2 = 25. Let λ be the line that is tangent to the circle at A. Find the angular size of minor arc AB, then find t

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Question 1158997: .Let A = (3,4) and B = (−3,4), which are both on the circle x2 +y2 = 25. Let λ be the line that is tangent to the circle at A. Find the angular size of minor arc AB, then find the size of the acute angle formed by λ and chord AB. Is there a predictable relation between the two numbers? Explain
Found 2 solutions by KMST, ikleyn:
Answer by KMST(5422) About Me  (Show Source):
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The circle x%5E2%2By%5E2=25 is the set of points at a distance of 5 from point O%280%2C0%29 .
In other words, it is the circle of radius 5 , centered at O%280%2C0%29 .
Chord AB is the segment of line y=4 connecting points A%283%2C4%29 and B%28-3%2C4%29 .
A line tangent to the circle at a certain point is perpendicular to the radius at that point,
so line green%28lambda%29 tangent to the circle at A must be perpendicular to radius OA .
The slope of radius OA is %284-0%29%2F%283-0%29=4%2F3 ,
so the slope of a line perpendicular to radius OA must be -1%2F%224+%2F+3%22=-3%2F4 .
The slope of line green%28lambda%29 , perpendicular to radius OA at A is the tangent of the angle red%28alpha%29 between the positive x-axis and green%28lambda%29 .
That is the same angle between the rightward extension of BA and green%28lambda%29 .
tan%28red%28alpha%29%29=-3%2F4=-0.75} --> red%28alpha%29=-36.9%5Eo%28rounded%29
The measures of the angles between line AB and green%28lambda%29 are approximately highlight%2836.9%5Eo%29 for the acute angle and 180%5Eo-36.9%5Eo=143.1%5Eo for the obtuse angle.

Answer by ikleyn(53996) About Me  (Show Source):
You can put this solution on YOUR website!
.
Let A = (3,4) and B = (−3,4), which are both on the circle x^2 + y^2 = 25. Let λ be the line that is tangent
to the circle at A.
(a) Find the angular size of minor arc AB,
(b) then find the size of the acute angle formed by λ and chord AB.
Is there a predictable relation between the two numbers? Explain
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 


Let O be the origin of the coordinate plane, so O is the center of the given circle.

Let C be the point C = (3,0).

Let alpha be the angle between radius OA and the x-axis.

Let beta be the angle between radius OA and the y-axis.


Then tan%28alpha%29 = AC%2FOC = 4%2F3 = 1.3333...,

alpha = arctan%284%2F3%29 = 53.13°, approximately.


Angle beta is the complement of alpha to 90°, so beta = 90°-53.13° = 36.87°, approximately.


Question (a) asks to find angle 2%2Abeta = 2 * 36.87° = 73.74°, approximately.


It is the answer to question (a): the minor arc AB has the angular measure of 73.74°, approximately.



To answer question (b), notice that the chord AB is parallel to the x-axis.

Therefore, the acute angle formed by λ and chord AB is the same as the acute angle formed by λ and the x-axis.


Since the line λ is perpendicular to the radius OA (according to a well-known property of a tangent line),
the acute angle formed by λ and the x-axis is the complement of angle alpha to 90°,
which is 90° - 53.13° = 36.87°.


So, the answer to question (b) is 36.87°.

Both questions are answered, and the problem is solved completely.