The minimum distance between any two points P1 and P2 while considering point P1 on one circle and p

Q: Wait, why do we use the center-to-center distance? Can't we just pick arbitrary points on the circles?

That is a fantastic question! If you pick arbitrary points, you are just finding the distance between two specific points, not the *minimum* distance. The geometry of a circle is perfectly symmetric. The shortest path between two circles must lie on the line that connects their centers because that line is the 'normal' to both circles. Any other path would be longer.

Q: What if the circles were intersecting? Would the formula $d - (r_1 + r_2)$ still work?

If the circles intersect, the distance between their centers $d$ will be less than the sum of their radii ($d < r_1 + r_2$). If you plug that into the formula $d - (r_1 + r_2)$, you would get a negative number, which doesn't make sense for a distance! In the case of intersection, the circles share points, so the minimum distance is simply $0$.

Q: Is there a case where the minimum distance is calculated differently?

Yes! If one circle is completely inside the other, the distance between centers $d$ plus the smaller radius $r_{small}$ will be less than the larger radius $r_{large}$. In that scenario, the minimum distance is $r_{large} - (d + r_{small})$. Always check the relative position of the circles first!

Question

The minimum distance between any two points $P_1$ and $P_2$ while considering point $P_1$ on one circle and point $P_2$ on the other circle for the given circles' equations $x^2 + y^2 - 10x - 10y + 41 = 0$ and $x^2 + y^2 - 24x - 10y + 160 = 0$ is ____.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, fellow traveler of the coordinate plane! Today, we are embarking on a journey to solve a classic problem in coordinate geometry. Imagine two planets floating in the vast, silent void of space.

We are given their equations, and our mission is to find the absolute shortest path between their surfaces. This isn't just about plugging numbers into a formula; it is about visualizing the geometry of the situation.

### Decoding the Equations

We are presented with two circles:

1. $C_1: x^2 + y^2 - 10x - 10y + 41 = 0$
2. $C_2: x^2 + y^2 - 24x - 10y + 160 = 0$

Before we can calculate any distances, we must strip away the algebraic disguise. We use the general equation of a circle: $x^2 + y^2 + 2gx + 2fy + c = 0$.

By comparing our given equations to this standard form, we can extract the vital statistics of each circle: the center $(h, k) = (-g, -f)$ and the radius $r = \sqrt{g^2 + f^2 - c}$.

For the first circle, $C_1$, we see that $2g = -10$ and $2f = -10$, which gives us $g = -5$ and $f = -5$. The constant $c$ is $41$.

Plugging these into our formulas, the center $C_1$ is $(5, 5)$, and the radius $r_1$ is calculated as follows:

$$r_1 = \sqrt{(-5)^2 + (-5)^2 - 41} = \sqrt{25 + 25 - 41} = \sqrt{9} = 3$$

We repeat this elegant dance for the second circle, $C_2$. Here, $2g = -24$ and $2f = -10$, so $g = -12$ and $f = -5$. The constant $c$ is $160$.

The center $C_2$ is $(12, 5)$, and the radius $r_2$ is calculated as follows:

$$r_2 = \sqrt{(-12)^2 + (-5)^2 - 160} = \sqrt{144 + 25 - 160} = \sqrt{9} = 3$$

### The Geometry of Distance

Now that we have our centers $C_1(5, 5)$ and $C_2(12, 5)$ and our radii $r_1 = 3$ and $r_2 = 3$, we can find the distance $d$ between the centers. Using the distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$, we get:

$$d = \sqrt{(12 - 5)^2 + (5 - 5)^2} = \sqrt{7^2 + 0^2} = 7$$

This is a beautiful moment. The $y$-coordinates are identical, meaning the line connecting the centers is perfectly horizontal. The distance between the centers is exactly $7$ units.

### The Final Leap

Before we declare victory, we must perform the 'Relative Position Check'—a crucial step in any JEE problem involving two circles. We compare the distance between the centers ($d = 7$) with the sum of the radii ($r_1 + r_2 = 3 + 3 = 6$).

Since $7 > 6$, we know with absolute certainty that these two circles are completely external to each other. They do not touch, and they do not overlap.

Because they are external, the shortest path between them lies along the line segment connecting their centers. To find this minimum distance, we take the total distance between the centers and subtract the radius of the first circle, and then subtract the radius of the second circle:

$$d_{min} = d - (r_1 + r_2)$$

$$d_{min} = 7 - (3 + 3) = 7 - 6 = 1$$

And there it is! The minimum distance between any point $P_1$ on the first circle and any point $P_2$ on the second circle is exactly **$1$ unit**. We have successfully navigated the coordinate plane, decoded the equations, and found the shortest path.

Enter Numerical Value:

Visualized Solution

The Two Circles

The General Equation of a Circle

Analyzing Circle 1

Center and Radius of Circle 1

Analyzing Circle 2

Center and Radius of Circle 2

Distance Between Centers

Calculating the Distance

Relative Position Check

The Minimum Distance Formula

Final Calculation

Summary and Takeaway

The Sigma Insight: Standard and General Equation of a Circle

Analyzing the Setup

Decoding the Equations

The Geometry of Distance

The Final Leap

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