Problem 35 - Algebra for JEE ADVANCED

Question

The number of ordered pairs of integers \((x, y)\) satisfying the equation \(x^{2}+6 x+y^{2}=4\) is a. 2 b. 8 c. 6 d. none of these

Step-by-Step Solution

Answer

The number of ordered pairs is 8.

Step 1: Understand the Equation

We are given the equation \(x^2 + 6x + y^2 = 4\). The goal is to find integer solutions for \((x, y)\) that satisfy this equation.

Step 2: Complete the Square

Rewrite the equation by completing the square for \(x\). This involves taking \(x^2 + 6x\) and writing it as \((x+3)^2 - 9\). The equation becomes \((x+3)^2 + y^2 = 13\).

Step 3: Recognize a Circle Equation

The equation \((x+3)^2 + y^2 = 13\) represents a circle centered at \((-3, 0)\) with a radius of \(\sqrt{13}\). We seek integer points on this circle.

Step 4: Identify Integer Points

For \((x+3)^2 + y^2 = 13\), \((x+3)\) and \(y\) must be integer values such that their sum squared is 13. Evaluate possible values:

Step 5: Evaluate Possible Values of \((x+3)^2\)

The integer squares on the circle with a radius \(\sqrt{13}\) are: - If \((x+3)^2 = 4\), \(y^2 = 9\) with solutions \((x+3, y) = (2, 3), (2, -3), (-2, 3), (-2, -3)\).- If \((x+3)^2 = 9\), \(y^2 = 4\) with solutions \((x+3, y) = (3, 2), (3, -2), (-3, 2), (-3, -2)\).Convert each \(x+3\) back to \(x\) by subtracting 3.

Step 6: List All Integer Values

For \((x+3, y) = (2, 3), (2, -3), (-2, 3), (-2, -3), (3, 2), (3, -2), (-3, 2), (-3, -2)\), converting to \(x\) gives - \((x, y) = (-1, 3), (-1, -3), (-5, 3), (-5, -3), (0, 2), (0, -2), (-6, 2), (-6, -2)\).

Step 7: Count the Number of Ordered Pairs

There are a total of 8 valid ordered pairs: 1. \((-1, 3)\) 2. \((-1, -3)\) 3. \((-5, 3)\) 4. \((-5, -3)\) 5. \((0, 2)\) 6. \((0, -2)\) 7. \((-6, 2)\) 8. \((-6, -2)\).

Key Concepts

circle equations

Circle equations are a fundamental part of geometry in mathematics. They represent all points located at a specific distance (the radius) from a particular point (the center). In the plane, the general 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.

In the context of our exercise, we transformed the equation \( x^2 + 6x + y^2 = 4 \) into \( (x+3)^2 + y^2 = 13 \). This tells us that the circle is centered at \((-3, 0)\) with a radius of \( \sqrt{13} \). Understanding circle equations helps us locate sets of points meeting specific criteria, like finding all integer points on a circle.

completing the square

Completing the square is a valuable technique used in algebra to transform quadratic equations into a form that's easier to work with. The purpose is to create a perfect square trinomial from a quadratic expression. This method simplifies the process of analyzing and solving quadratic equations. For the given equation \( x^2 + 6x + y^2 = 4 \), we focused on the part \( x^2 + 6x \). To complete the square:

• Take the coefficient of \( x \), which is \( 6 \), and divide it by \( 2 \) getting \( 3 \).
• Square the result to get \( 9 \).
• Reformulate \( x^2 + 6x \) as \((x + 3)^2 - 9\).

So, \( x^2 + 6x \) becomes \((x + 3)^2 - 9\), giving us an expression directly in the circle equation form.

ordered pairs of integers

An ordered pair of integers \((x, y)\) is a fundamental concept in coordinate geometry. It represents the precise location of a point on a plane, where \(x\) and \(y\) are integer values (whole numbers without fractions).In solving for these pairs, we look for combinations of \(x\) and \(y\) that satisfy a certain equation or condition.
In our equation \((x+3)^2 + y^2 = 13\), we identified integer solutions by first evaluating potential integer values of \((x+3)^2\) which can be \(4\) or \(9\), leading us to corresponding \(y^2\) values of \(9\) and \(4\), respectively.
Each integer solution for \(x+3\) and \(y\) allows us to find valid \(x\) values, resulting in a total of eight ordered pairs that satisfy the circle equation. This visualization on the integer plane helps us see that while continuous, a circle will have discrete solutions when restricted to integers.

JEE Advanced mathematics

JEE Advanced mathematics is a section of the Joint Entrance Examination, an academic aptitude test in India designed to assess understanding at a high level. It evaluates conceptual clarity and analytical skills, often through complex problem-solving exercises related to algebra, geometry, and calculus. A typical question, like the problem given, challenges students to use their knowledge in creative ways, such as through transformation techniques like completing the square or finding integer solutions to circle equations.
Preparation for JEE Advanced mathematics requires consistent practice with challenging problems and a deep understanding of core concepts. Students should be comfortable with algebraic manipulations, recognizing geometric shapes through equations, and counting solutions that satisfy multiple conditions, all of which are key skills tested in the exam. These questions not only test computational skill but also the ability to think logically and reach conclusions based on given mathematical implications.