Problem 131 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(C\) be the circle with centre at \((1,1)\) and with radius 1\. If \(T\) is the circle centered at \((0, y)\), passing through origin and touching the circle \(C\) externally, then the radius of \(T\) is equal to [2014] (A) \(\begin{array}{ll}\frac{\sqrt{3}}{\sqrt{2}} & \text { (B) } \frac{\sqrt{3}}{2}\end{array}\) (C) \(\frac{1}{2}\) (D) \(\frac{1}{4}\)

Step-by-Step Solution

Answer

The radius of circle \(T\) is \(\frac{1}{4}\).

Step 1: Determine Radius of Circle C

The radius of circle \(C\) is given as 1 since it is a standard circle with center \((1,1)\). This means the equation of circle \(C\) is \((x - 1)^2 + (y - 1)^2 = 1^2\).

Step 2: Determine the Characteristics of Circle T

Circle \(T\) is centered at \((0, y)\), passes through the origin \((0,0)\), and its radius extends from its center to the origin. Hence, the radius \(r_t\) of circle \(T\) is \(\sqrt{(0-0)^2 + (y-0)^2} = |y|\).

Step 3: Calculate the Distance Between Circle Centres

The distance between the centers of \(C\) and \(T\) is \(\sqrt{(1-0)^2 + (1-y)^2} = \sqrt{1 + (1-y)^2}\).

Step 4: Determine Equation for External Tangency

Since circle \(T\) touches circle \(C\) externally, the sum of their radii equals the distance between their centers. Thus, we have \(r_t + 1 = \sqrt{1 + (1-y)^2}\). Substitute \(r_t = |y|\): \(|y| + 1 = \sqrt{2 - 2y + y^2}\).

Step 5: Simplify and Solve for Radius y

Square both sides to remove the square root: \((|y| + 1)^2 = 2 - 2y + y^2\)Expanding both sides:\(y^2 + 2|y| + 1 = 2 - 2y + y^2\)Simplifying for \(|y|\):\(2|y| + 1 = 2 - 2y\)From \(|y| = y\), solve \(2y + 1 = 2 - 2y\)This gives \(4y = 1\) or \(y = \frac{1}{4}\).

Step 6: Verify and Check Result

Since \(y\) is positive based on our assumption \(|y| = y\), the correct solution is \(y = \frac{1}{4}\), which is the radius of circle \(T\).

Key Concepts

external tangency

When two circles are externally tangent, they touch at exactly one point, but they do not intersect or overlap. This is a crucial property in circle geometry. External tangency means that the distance between the centers of the two circles is exactly equal to the sum of their radii.
For example, if you have two circles, circle C with radius 1 and circle T with a certain radius, the distance between their centers needs to equal the sum of these two radii for them to be externally tangent.
This property allows for precise calculations, ensuring that the circles maintain their structural properties without overlapping. In practical scenarios, understanding external tangency can help with designing circular objects that need to fit perfectly within a given space.

circle equations

In circle geometry, the equation of a circle is fundamental. The standard form of a circle's equation with center at \(h, k\) and radius \(r\) is \((x - h)^2 + (y - k)^2 = r^2\).
To illustrate, consider circle C with a center at (1,1) and radius 1. The equation comes out as \( (x - 1)^2 + (y - 1)^2 = 1^2 \), which simplifies to \( (x - 1)^2 + (y - 1)^2 = 1 \). This helps determine every point that lies on the circle.
For the second circle, circle T, things get interesting because its center is positioned at (0, y), and it must pass through the origin (0,0). Thus, the radius is \|y|\, leading to \( (x - 0)^2 + (y - y)^2 = |y|^2 \). This basic setup allows us to explore the relationship between the circles and solve problems such as finding intersections or ensuring tangency.

radius determination

Determining the radius is often crucial, especially in problems involving tangency or intersections. For circle C, the radius is straightforward, given as 1.

• This simplifies its equation and concepts revolving around it.
  
• For circle T, though, radius determination requires additional steps. Centered at (0, y), the radius is \|y|\, which automatically defines it based on the coordinates.
  

The problem of determination becomes slightly complex due to the external tangency condition, where the total distance between the circles' centers must match the radii sum. Expressed in equation form, this becomes \(|y| + 1 = \sqrt{2 - 2y + y^2}\).
Once simplified to \(2y + 1 = 2 - 2y\), and since \|y| = y\, solving gives y = \(\frac{1}{4}\). Thus, the radius of circle T is perfectly determined and verified through logical steps.