Problem 9 - Algebra for JEE ADVANCED

Question

Let \(f(n)\) be the number of regions in which \(n\) coplanar circles can divide the plane. If it is known that each pair of circles intersect in two different point and no three of them have common point of intersection, then a. \(f(20)=382\) b. \(f(n)\) is always an even number c. \(f^{-1}(92)=10\) d. \(f(n)\) can be odd

Step-by-Step Solution

Answer

a, b, and c are true; d is false.

Step 1: Understanding the Problem

We are given multiple statements about a function \( f(n) \) that determines the number of regions formed by \( n \) coplanar circles, with each pair intersecting at exactly two points and no three sharing a common intersection point. We need to evaluate the truth of these statements.

Step 2: Identifying Patterns or Formulas

When each new circle is added, it intersects the existing \( n-1 \) circles twice, adding \( 2(n-1) \) new regions. The total number of regions formed by \( n \) circles is derived by using the formula:\[ f(n) = f(n-1) + 2(n-1) \] Starting with \( f(1) = 2 \) (1 circle divides the plane into 2 regions), we can incrementally build up regions for additional circles.

Step 3: Calculate and Confirm f(20)

Compute \( f(20) \) recursively or by discovering a summation pattern derived:1. Start with \( f(1) = 2 \)2. Each additional circle adds regions according to: \[ f(n) = 2 + 2 \times 0 + 2 \times 1 + 2 \times 2 + \,\ldots\, + 2 \times (n-1) \]Summing simplifies to: \[ f(n) = n^2 - n + 2 \]Substituting \( n=20 \) gives us:\[ f(20) = 20^2 - 20 + 2 = 382 \]which confirms statement (a) is true.

Step 4: Check if f(n) can be Odd or Always Even

The derived formula gives:\[ f(n) = n^2 - n + 2 \]which can be rewritten as: \[ f(n) = (n(n-1) + 2) \]which means \( f(n) \) is always even, since the expression \( n(n-1) \) is a product of two consecutive integers (one of which is even), and adding 2 keeps it even. Thus, statement (d) is false and (b) is true.

Step 5: Calculate f^{-1}(92) and Validate

We need to find \( n \) such that \( f(n) = 92 \) using our formula:\[ n^2 - n + 2 = 92 \]Simplifying it, we have:\[ n^2 - n - 90 = 0 \]Use the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -1 \), and \( c = -90 \), gives:\[ n = \frac{1 \pm \sqrt{1 + 360}}{2} = \frac{1 \pm 19}{2} \]Giving us \( n = 10 \) or \( n = -9 \). Thus, \( f^{-1}(92) = 10 \) valid asserts (c) as true.

Key Concepts

Coplanar Circles

Coplanar circles are circles that lie flat on the same plane, like rings on a tabletop. When considering multiple coplanar circles, their interactions can create complex patterns of overlapping regions.
This is a key concept in algebraic geometry, which studies shapes and their properties based on algebraic systems.
In our problem, each pair of circles intersects in two distinct points. This regularity means every two circles create new regions as they intersect, but no three circles intersect at a single point, which simplifies the potential complexity.

Regions in Plane

When circles intersect in a plane, they divide it into several distinct areas. Each new circle added increases the number of these regions exponentially. This process starts with one circle dividing the plane into two parts and expands as more circles join and intersect.
In the function we are exploring, given by \( f(n) \), it represents how many separate regions these circles create in total.
Initially, one circle divides the plane into two regions. Adding a second circle, which intersects the first at two points, divides the plane into four regions. More circles keep increasing this number based on their intersections, following a logical pattern: each additional circle introduces new intersections, splitting existing regions further.

Quadratic Equations

Quadratic equations are equations of the form \( ax^2 + bx + c = 0 \). They are key in many mathematical calculations, especially in algebraic geometry. In relation to our problem, solving a quadratic equation helps us determine the number of circles needed to reach a specified number of plane regions.
The quadratic formula is a handy tool for this, allowing us to find solutions for \( n \) when a function like \( f(n) = n^2 - n + 2 \) is set to a particular value.
For instance, calculating \( f(n) = 92 \) involved rearranging the function into the standard quadratic form and using \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the possible values of \( n \).

Intersection Points

Intersection points occur where two circles meet in a plane. In our specific case of coplanar circles, each pair interacts at exactly two points. These points of intersection are fundamental because they help define the regions created by the array of circles.
Every additional intersection adds complexity and further divides the plane, resulting in more regions to consider. Given our assumption that no three circles meet at the same point, managing the intersections becomes a straightforward task of calculating how new circles contribute by pairing points.
This simplification helps in forming a pattern with the function \( f(n) \), allowing for easy calculation of total regions by simply accounting for each circle's contribution through its intersections.