Limit problem involving two circles and a line

In summary, the problem involves finding the equation of PQ using the limiting position of R, (4,0). The method used involves finding the coordinates of Q and then using the equation for a line to find the coordinates of R as a function of the radius r. The final step is to calculate the limit of this function as r approaches 0.
  • #1
ChiralSuperfields
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Homework Statement
Please see below
Relevant Equations
Please see below
For this problem,
1681963648890.png

The limiting position of R is (4,0). However, I am trying to solve this problem using a method that is different to the solutions. So far I have got,

##C_1: (x - 1)^2 + y^2 = 1##
##C_2: x^2 + y^2 = r^2##

To find the equation of PQ,
## P(0,r) ## and ##R(R,0) ##
## y = \frac{r(x - R)}{-R} ##
Then solve for ## R ## to get,

##R = \frac{rx}{r - y}##
##R = \frac{rx}{r - \sqrt{r^2 + x^2}} ##
##R = \lim_{r \rightarrow 0^+} \frac{rx}{r - \sqrt{r^2 + x^2}} = 0 ##

Can someone please give guidance to what I have done wrong?

Many thanks!
 
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  • #2
What did you get as coordinates for ##Q##?
 
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  • #3
fresh_42 said:
What did you get as coordinates for ##Q##?
Thank you for your reply @fresh_42! I missed up finding the coordinates of ##Q## in my original attempt.

But according to the solutions,
1681980177755.png

However, I am curious how to solve this problem without using Q's coordinates.

Many thanks!
 
  • #4
You need the equation for the line ##\overline{PQ}## which is defined by ##Q## so you definitely need the coordinates for ##Q## somehow; if not explicitly then implicitly. Say the straight is ##y=mx+b.## Then ##R## has the coordinates ##R=(-\frac{b}{m},0).## We finally need to solve ##\lim_{r \to 0^+} \frac{-b}{m}.##

I do not see how to get there without ##m## and ##b## that are determined by ##P## and ##Q.##
 
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  • #5
Hint for the limit calculation:

a) Show that the quotient is greater than ##2## for ##0<r < 2.##
b) Set the quotient equal to ##L## and solve for ##r^2.##
c) Show what happens to ##L## if ##r \rightarrow 0.##
 
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  • #6
fresh_42 said:
You need the equation for the line ##\overline{PQ}## which is defined by ##Q## so you definitely need the coordinates for ##Q## somehow; if not explicitly then implicitly. Say the straight is ##y=mx+b.## Then ##R## has the coordinates ##R=(-\frac{b}{m},0).## We finally need to solve ##\lim_{r \to 0^+} \frac{-b}{m}.##

I do not see how to get there without ##m## and ##b## that are determined by ##P## and ##Q.##
Thank you for your replies @fresh_42!

I think part of my confusion is that I think that the line ##\overline{PQ} = \overline{PR}##. Do you please know why they are different?

I also think I wrongly wrote ##R## as ##R(R,0)##, because I was thinking that the distance from the origin to point ##R## is ##R##. I now think that is incorrect because the problem only specifies the point as ##R##. Do you please know whether that is the only reason why we cannot assume that ##R(R,0)##?

Many thanks!
 
  • #7
fresh_42 said:
You need the equation for the line ##\overline{PQ}## which is defined by ##Q## so you definitely need the coordinates for ##Q## somehow; if not explicitly then implicitly. Say the straight is ##y=mx+b.## Then ##R## has the coordinates ##R=(-\frac{b}{m},0).## We finally need to solve ##\lim_{r \to 0^+} \frac{-b}{m}.##

I do not see how to get there without ##m## and ##b## that are determined by ##P## and ##Q.##
Sorry, do you please know what it means to find the coordinates of ##Q## explicitly or implicitly?

Would finding the coordinates of ##Q## explicitly translate to find ##Q## as a function of some variables, for example ##R=(-\frac{b}{m},0)## like you wrote?

And finding coordinates of ##Q## implicitly would translate to ##Q## as a function of some other variables?

Many thanks!
 
  • #8
ChiralSuperfields said:
Thank you for your replies @fresh_42!

I think part of my confusion is that I think that the line ##\overline{PQ} = \overline{PR}##. Do you please know why they are different?
They are not. ##P## and ##R## change when the left circle shrinks. They, and ##Q##, depend on its radius ##r.##
ChiralSuperfields said:
I also think I wrongly wrote ##R## as ##R(R,0)##, because I was thinking that the distance from the origin to point ##R## is ##R##.
That is both true. ##R=(R,0).## A better notation would be ##R=(R(r),0)## as the coordinate changes with ##r.##
ChiralSuperfields said:
I now think that is incorrect because the problem only specifies the point as ##R##. Do you please know whether that is the only reason why we cannot assume that ##R(R,0)##?

Many thanks!

You are almost done. You correctly calculated ##Q=\left(\dfrac{r^2}{2}\, , \,\dfrac{r}{2}\sqrt{4-r^2}\right).##

The rest is not so difficult. The equation for a line given by points ##A=(x_a,y_a)## and ##B=(x_b,y_b)## is
$$
\dfrac{y-y_a}{x-x_a}=\dfrac{y_b-y_a}{x_b-x_a}
$$
So calculate this expression with ##A=P## and ##B=Q,## bring it into the form ##y=m\cdot x+ b## and calculate ##R=(x_0,0)=(x_0,m\cdot x_0+b),## i.e. 0=m\cdot x_0+b## or ##x_0=-\dfrac{b}{m}=R(r).##

This gives you the ##x##-coordinate of the point ##R## as a function of ##r,## the radius of the left circle.
Then proceed along the lines in post #5 in order to find the limit mentioned in post #4.
 
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  • #9
ChiralSuperfields said:
Sorry, do you please know what it means to find the coordinates of ##Q## explicitly or implicitly?
You have computed the coordinates of ##Q##. Correctly calculated them! So you can work with these formulas, which would be an explicit use.

We need the information hidden in the position of ##Q## because it defines our problem. We need ##Q.## If you do not want to use the coordinates, the result of your calculation, then you have to use ##Q## otherwise. That would be an implicit use since you had to use the definition of ##Q## without calculating it. Such usage is called implicit use since you avoid the explicit solution. I have no idea how to do that and consider it unnecessary especially as you do have the coordinates already. I just answered your idea of doing it without knowing the coordinates.

E.g. The point ##R## is implicitly given. We know how to construct it since we have a description of the algorithm, but we do not know the explicit coordinates, yet.

Calculate them next: ##R=\left(-\dfrac{b}{m},0\right).## What are ##m## and ##b## as a function of ##r##?


ChiralSuperfields said:
Would finding the coordinates of ##Q## explicitly translate to find ##Q## as a function of some variables, for example ##R=(-\frac{b}{m},0)## like you wrote?

And finding coordinates of ##Q## implicitly would translate to ##Q## as a function of some other variables?

Many thanks!
 
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