Circle Centered At the Origin

[nycmathguy]

Set 1.6
Question 50

See attachment

The following figure shows a circle centered at the origin and a line that is tangent to the circle at the point (3, -4).

 

[MathLover1]

a)
first find radius
r= distance from (0,0) to (3,-4)

r=

r=

r=5
The distance is 5.


b)
intercepts of the tangent line:

since radius is perpendicular to the tangent line, find equation of both lines

the line that passes through origin and (3,-4) has a slope
(-4-0)/(3-0)=-4/3
equation of the line containing radius is

y=-(4/3)x


then, the slope of perpendicular line will be: -1/(-4/3)=3/4

use point slope formula to find equation of the tangent line

y-y[1]=m(x-x[1])
y-(-4)=(3/4)(x-3)
y+4=(3/4)x-(3/4)*3
y+4=(3/4)x-9/4
y=(3/4)x-9/4-4
y=(3/4)x-9/4-16/4
y=(3/4)x-25/4

y-intercept is at
y=(3/4)0-25/4
y=-25/4=>y-intercept is at (0,-25/4)

x-intercept is at
0=(3/4)x-25/4
(3/4)x=25/4
x=(25/4)/(3/4)
x=25/3=>x-intercept is at (25/3,0)


c) the length of the portion of the tangent line in quadrant IV is equal to the distance between x and y-intercepts

(25/3,0) and (0,-25/4)

d=

d=

d=125/12

d=10.4

 

[nycmathguy]

a)
first find radius
r= distance from (0,0) to (3,-4)

r=

r=View attachment 190

r=5
The distance is 5.


b)
intercepts of the tangent line:

since radius is perpendicular to the tangent line, find equation of both lines

the line that passes through origin and (3,-4) has a slope
(-4-0)/(3-0)=-4/3
equation of the line containing radius is

y=-(4/3)x


then, the slope of perpendicular line will be: -1/(-4/3)=3/4

use point slope formula to find equation of the tangent line

y-y[1]=m(x-x[1])
y-(-4)=(3/4)(x-3)
y+4=(3/4)x-(3/4)*3
y+4=(3/4)x-9/4
y=(3/4)x-9/4-4
y=(3/4)x-9/4-16/4
y=(3/4)x-25/4

y-intercept is at
y=(3/4)0-25/4
y=-25/4=>y-intercept is at (0,-25/4)

x-intercept is at
0=(3/4)x-25/4
(3/4)x=25/4
x=(25/4)/(3/4)
x=25/3=>x-intercept is at (25/3,0)


c) the length of the portion of the tangent line in quadrant IV is equal to the distance between x and y-intercepts

(25/3,0) and (0,-25/4)

d=View attachment 191


d=View attachment 192


d=125/12

d=10.4

You are truly a mathematical genius. What amazes me is the fact that you saw this thread for the first time and knew precisely what to do. If I learn this problem in class, I can find the solution. However, I never saw this question before and David Cohen does not explain or hint how it is calculated. Like most math professors, they expect students to know what to do when facing a problem they have not been taught in class based on the chapter lesson. It takes a gifted person to see a question for the first time and find the answer.