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
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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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Related to Limit problem involving two circles and a line

What is a limit problem involving two circles and a line?

A limit problem involving two circles and a line typically refers to a mathematical scenario where you need to find the behavior of a function or a variable as it approaches a certain point, often where the two circles and the line interact. This can involve finding the limit of distances, angles, or other geometric properties as one or more parameters change.

How do you set up a limit problem with two circles and a line?

To set up a limit problem with two circles and a line, you first need to define the equations of the circles and the line. For example, if the circles are centered at (h1, k1) and (h2, k2) with radii r1 and r2, their equations are (x-h1)^2 + (y-k1)^2 = r1^2 and (x-h2)^2 + (y-k2)^2 = r2^2. The line can be defined by an equation like y = mx + b. You then formulate the problem by identifying the variable that approaches a limit, such as the distance between the circles or the angle between the line and a tangent to one of the circles.

What are common techniques to solve these limit problems?

Common techniques to solve limit problems involving two circles and a line include algebraic manipulation, geometric interpretations, and the use of calculus, particularly differentiation and L'Hôpital's Rule. In some cases, you might also use polar coordinates or parametric equations to simplify the problem.

Can you provide an example of a limit problem involving two circles and a line?

Sure! Consider two circles with equations (x-1)^2 + y^2 = 1 and (x-3)^2 + y^2 = 1, and a line y = mx. Find the limit of the distance between the points of intersection of the line with each circle as m approaches 0. As m approaches 0, the line becomes horizontal, and the points of intersection will approach (1,0) and (3,0), respectively. The distance between these points is 2, so the limit is 2.

What are some real-world applications of these limit problems?

Real-world applications of limit problems involving two circles and a line can be found in fields like engineering, physics, and computer graphics. For example, in robotics, understanding the limits of movement paths can help in designing better algorithms for navigation. In physics, such problems can help in analyzing particle trajectories or wave propagation. In computer graphics, they are useful in rendering scenes with curves and lines accurately.

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