Problem 6 - A Complete Resource Book in Mathematics for JEE Main

Question

If the equations of four circles are \((x \pm 4)^{2}+(y \pm 4)^{2}\) \(=4^{2}\), then the radius of the smallest circle touching all the four circles is (A) \(4(\sqrt{2}+1)\) (B) \(4(\sqrt{2}-1)\) (C) \(2(\sqrt{2}-1)\) (D) none of these

Step-by-Step Solution

Answer

The radius of the smallest circle is (B) \(4(\sqrt{2}-1)\).

Step 1: Understand the positions of the circles

The given equations represent four circles centered at \((4, 4), (-4, 4), (4, -4), (-4, -4)\) all with radius 4. Visualizing the positions, these centers form a square of side 8 centered at the origin (0,0).

Step 2: Determine the distance to the new circle's center

To find the smallest circle touching all four circles, consider the center at the origin with radius extending to the nearest point of each circle. Distance from the origin (0,0) to any given center like (4,4) is calculated by \[\text{distance} = \sqrt{4^2 + 4^2} = 4\sqrt{2}\].

Step 3: Calculate the required radius

The circle we want must reach a distance just enough to intersect the other circles' radii, meaning the new circle's radius is the calculated distance minus the radius of the given circles: \[\text{radius} = 4\sqrt{2} - 4\]. Simplifying gives the smallest radius as \[4(\sqrt{2} - 1)\].

Step 4: Match the answer to the options

From the simplified radius, the choice that matches is (B) \(4(\sqrt{2}-1)\).

Key Concepts

Radius Calculation

Understanding how to calculate the radius of a circle is a fundamental skill in circle geometry. In this exercise, we are introduced to the concept by examining circles with the standard form of the circle equation: \[ (x - h)^2 + (y - k)^2 = r^2 \] where

• \((h, k)\) is the center of the circle
• \(r\) is the radius.

For the given problem, the equation \( (x \pm 4)^2 + (y \pm 4)^2 = 4^2 \) already provides the clue that the radius \(r\) is \(4\). When solving more complex problems where circles interact, understanding how to calculate or adjust this radius is key. In our situation, we need to find the smallest circle that touches all four given circles, hinting at dynamic radius calculations. By understanding the fixed radius of our four initial circles, we proceed to determine a new radius—one that's not part of the initial framework but crucial for the next steps.

Square Geometry

Square geometry finds its application in identifying and analyzing patterns when dealing with multiple circles’ placements. Here, the circle centers

• \((4, 4), (-4, 4), (4, -4), (-4, -4)\)

form a geometric structure: a square. Each point represents a corner of this square, which directly aligns with the known concepts of square properties. Each side of this square measures 8 units, calculated using the distance formula between the points (e.g., from \((4, 4)\) to \((4, -4)\)).
Identifying this square helps us quantify relationships, like where a new circle’s center might be. Frequently, square geometry ensures that we can utilize right-angled triangle properties and the Pythagorean theorem to successfully locate other important measurements or centers.

Coordinates of a Circle Center

Locating the circle centers involves interpreting each given equation to find

• \((h, k)\), the coords of the centers and
• how all relate to each other spatially.

From the equations \((x \pm 4)^{2}+(y \pm 4)^{2}=4^{2}\), each reveals either an addition or subtraction from \(x\) and \(y\), positioning the centers symmetrically around the origin. The strategic placement of \((4, 4), (-4, 4), (4, -4), (-4, -4)\) centers forms a square about the origin, a point \((0, 0)\).
To solve the problem, the new circle we seek is also centered at \((0, 0)\). This uniform distribution aids in making decisions about extending or adjusting radius values based on calculated distances. Each step requires a keen interpretation of how coordinates affect the circle's placement in relation with others—a valuable insight for geometrical problem-solving.