Problem 6 - A Complete Resource Book in Mathematics for JEE Main 2017

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 is \(4(\sqrt{2}-1)\), option B.

Step 1: Identify Center and Radius of the Circles

The given equations of the four circles are 1. \((x + 4)^2 + (y + 4)^2 = 4^2\), 2. \((x + 4)^2 + (y - 4)^2 = 4^2\), 3. \((x - 4)^2 + (y + 4)^2 = 4^2\), 4. \((x - 4)^2 + (y - 4)^2 = 4^2\).All these circles have a radius of 4 and their centers are \((-4, -4)\), \((-4, 4)\), \((4, -4)\), and \((4, 4)\) respectively.

Step 2: Determine Distance Between Centers

Calculate the distance between these centers to find how they are positioned. For example, the distance between \((4, 4)\) and \((-4, -4)\) is calculated using the distance formula:\[\sqrt{(4 - (-4))^2 + (4 - (-4))^2} = \sqrt{8^2 + 8^2} = \sqrt{128} = 8\sqrt{2}\].Similarly, all neighboring centers have distances of 8 apart.

Step 3: Determine Center and Radius of the Circle Touching All

To find the radius of a circle that touches all four given circles, note that this new circle's center will be equidistant from the four centers.Thus, its center can be calculated as the average of the coordinates of the centers of the circles:\((x, y) = \left(\frac{-4 + (-4) + 4 + 4}{4}, \frac{-4 + 4 + (-4) + 4}{4}\right) = (0, 0)\).The distance from (0, 0) to any of the circle centers is \(\sqrt{0^2 + 0^2} = 0\). Therefore,the distance from (0, 0) to any of the circle centers is \(\sqrt{16 + 16} = 4\sqrt{2}\).However, since these circles already have a radius of 4, the required distance (and thus radius of the circle touching all four) is \(4\sqrt{2} - 4\).

Step 4: Compute Simplified Radius Expression

Thus the radius of the circle that surrounds and touches all four circles is given by:\[R = 4(\sqrt{2}-1).\]This expression represents the effective radius of the circle which just touches all four smaller circles from the outside.

Key Concepts

Circle Equations

Understanding circle equations is key in solving problems related to circle geometry. A standard form of a circle equation is \[(x - h)^2 + (y - k)^2 = r^2\] where

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

Recognizing this form helps identify the geometric attributes of a circle from its equation. When equations are provided with terms \((x+4)^2\) or \((y-4)^2\), they indicate shifts from the origin to find the center. For instance, \((x + 4)^2 + (y + 4)^2 = 16\) has a center at \((-4, -4)\) and a radius of \(4\).Identifying these components helps you locate and understand the position of the circle in a coordinate plane.

Radius Calculation

The radius of a circle is the distance from its center to any point on the circle itself. Given the circle equation \((x - h)^2 + (y - k)^2 = r^2\), the radius is simply the square root of the right side, \(r\). In our exercise problems, we were given \((x \pm 4)^2 + (y \pm 4)^2 = 4^2\), which reveals that the radius \(r = 4\).Calculating the radius in contextual problems may involve understanding how distances change relative to new circles. For example, if a new circle is configured to "touch" another circle externally or internally, knowing adjacent distances can help you derive the radius for positioning or for size calculations.

Distance Formula

The distance formula is essential to understand the geometrical relationship between two points in a plane. The formula is\[\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\], where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.This formula allows you to calculate the distance between any two points, including the centers of circles.In the exercise, the centers of the given circles are \((-4, -4)\), \((-4, 4)\), \((4, -4)\), and \((4, 4)\). Calculating the distance between these points shows how they are positioned relative to one another. For example, the distance between \((4, 4)\) and \((-4, -4)\) is \(8\sqrt{2}\). This knowledge was vital in determining the settings for a circle that touches all four circles.

Circle Center

The center of a circle is the pivot point around which the entire circle rotates. In typical circle equations like \((x - h)^2 + (y - k)^2 = r^2\), \((h, k)\) signifies the center.In our problem, the centers were easy to identify due to the perfectly symmetric nature of the circle equations given for the four smaller circles. Furthermore, to find a center that is equidistant from multiple known points, like in the case of the larger circle touching all four smaller circles from the exercise, you use arithmetic averages. Hence, the center for the new circle turned out to be \((0, 0)\).Locating the center is a step that greatly simplifies understanding and solving geometry problems related to circles, especially when calculating or conceptualizing relationships between multiple circles or when working on tangent problems.