# Calculating Minimum Distance p+q

• Callisto
In summary, the conversation discusses determining the minimum distance between an object and image for a given focal length using the equation 1/p + 1/q = 1/f. Tips are given on how to solve the equation and graph it, and it is noted that the relationship is a hyperbola. The process for finding the minimum value is explained and it is concluded that the minimum distance is p+q = 4f. The exclusion of 0 from the problem is also discussed and it is determined that 2f is the minimum distance since 1/p=0 when the object is at infinity.
Callisto
Hi Peaple :!)

How do you determine the minimum distance p+q between the object and image for given focal length given

1/p + 1/q = 1/f

this may seem trivial, but i can't figure it out.

Callisto

Arent p and q distances...? Just add them.

If your saying given a fixed focal length, find the relationship between p and q, that case

$$\frac{1}{p} + \frac{1}{q} = k$$

Solve it for either p or q (the result is the same) and graph it. I believe its a logarithmic relation, but I'm not sure.

What math tools do you have at your disposal?

Actually the thin lens relationship is a hyperbola.

I should know that, my brother did a presentation on it a week ago.

solving 1/p+1/p=1/f for p
i get
-qf/(f-q)
now i substitute this into p+q, then i get -q^2/(f-q).
now if i use calculus to find the minimum value i differentiate -q^2/(f-q) and let it equal zero so
d/dq= -q(2f-q)/(f-q)^2 =0 when q = 0 or 2f

Repeating this process for p then p = 0 or 2f
so the minimum distance is
p+q=4f. why is it not zero

Callisto

Looks fine to me. You might want to check that 2f is a minimum (not a maximum or point of inflection), to be thorough. Looking at the lens equation itself do you see why the 0 solutions are inadmissible?

Last edited:
1/0+1/0=0 , 0 doesn't = 1/f
therefore 0's are excluded from the problem, correct?
Callisto

1/0 + 1/0 is 0? Are you sure?

1/0+1/0=1/0, 1/0 doesn't = 1/f
is this why 0's are excluded?

1/0 is undefined, as is 1/0+1/0, and no other arithmetic with these quantities is defined either, so if p or q were 0, you could not rearrange the equation the way that you did (without first defining a new arthmetic). In order for the lens equation to make sense with usual definitions, the assumption p, q not zero must be made.

Thanks
2f must be minimum since if the object p is at infinity then 1/p=0 so q = f for an object at infinity.

## 1. What is the purpose of calculating minimum distance p+q?

The purpose of calculating minimum distance p+q is to determine the shortest distance between two points in a given space. This is useful in various fields such as engineering, physics, and computer science where finding the shortest path or distance is essential for solving problems.

## 2. How is minimum distance p+q calculated?

Minimum distance p+q is calculated using the Pythagorean theorem, which states that the square of the hypotenuse (the longest side) of a right triangle is equal to the sum of the squares of the other two sides. In this case, p and q represent the two sides of the triangle, and the square root of their sum gives the minimum distance.

## 3. What are some real-life applications of calculating minimum distance p+q?

Calculating minimum distance p+q has various real-life applications, such as determining the shortest route for a delivery driver, finding the closest hospital in case of an emergency, or calculating the shortest distance between two cities for travel planning. It is also used in navigation systems, robotics, and satellite imagery analysis.

## 4. Can minimum distance p+q be calculated in any space?

Yes, minimum distance p+q can be calculated in any space as long as there are two defined points in that space. This includes two-dimensional and three-dimensional spaces, as well as spaces with more complex dimensions, such as in higher mathematics or physics.

## 5. Are there any limitations to calculating minimum distance p+q?

One limitation of calculating minimum distance p+q is that it assumes a straight line path between the two points. In reality, obstacles or barriers may exist, making the actual minimum distance longer than the calculated value. Additionally, this method only works for finding the shortest distance between two points and may not be applicable for finding the shortest path between multiple points.

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