Question

In a triangle \(\mathrm{ABC}\), if \((\mathrm{a}+\mathrm{b}+\mathrm{c})(\mathrm{b}+\mathrm{c}-\mathrm{a})=\lambda \mathrm{bc}\), then (a) \(\lambda<0\) (b) \(\lambda>6\) (c) \(0<\lambda<4\) (d) \(\lambda>4\)

Short answer

Answer: (c) \(0<\lambda<4\).

Step-by-step solution

Apply the Triangle Inequality Theorem

According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For triangle ABC, this means: 1. \(a+b>c\) 2. \(a+c>b\) 3. \(b+c>a\)

Use the constraints to simplify the equations

From the triangle inequality theorem, we can obtain the following inequalities: 1. \(a+b-c>0 \implies c0 \implies b0 \implies a<b+c\) Now, we will substitute these inequalities into the given equation \((a+b+c)(b+c-a) = \lambda bc\).

Substitute the inequalities and find the range of \(\lambda\)

We can rewrite the equation as follows: \(\mathrm{c}(\mathrm{a}+\mathrm{b})+\mathrm{a}(\mathrm{b}-\mathrm{c}) = \mathrm{a}\mathrm{b}\mathrm{c}(1 - \frac{1}{\lambda})\) Let's take the case of \((a, b, c) = (3, 4, 5)\) as an example: \(5(3+4)+3(4-5) = 3\cdot 4\cdot 5(1 - \frac{1}{\lambda})\) \(5\cdot 7 - 3 = 60(1 - \frac{1}{\lambda})\) \(32 = 60(1 - \frac{1}{\lambda})\) Now, we can solve for \(\lambda\): \(\frac{1}{\lambda} = 1 - \frac{32}{60}\) \(\frac{1}{\lambda} = \frac{28}{60}\) \(\lambda = \frac{60}{28} = \frac{15}{7}\) Since we have found one possible value for \(\lambda\), we can safely say that it is not less than 0 (option a). Now, we will investigate the options b, c, and d.

Check remaining options

To check options b, c, and d, we need to see whether we can find a value for \(\lambda\) that is greater than \(6, 4,\) or within the range of \(0\) to \(4\). By setting a very large value for c: \(5\cdot 1001 - 3 = 3\cdot 4\cdot 1001(1 - \frac{1}{\lambda})\) Manipulating the equation, we get: \(\lambda = \frac{1}{1-\frac{5003}{12012}} = \frac{12012}{5003-12012} = -\frac{12012}{7009}\) For this example, we found \(\lambda\) to be negative, which contradicts option a. However, as we have found a positive value for \(\lambda\) before, this simply shows that there is no fixed value for \(\lambda\) and it can have a wide range, including negative values. Since we have found an example where \(\lambda\) is within the range of \(0\) to \(4\), we can conclude that option c is correct. So the answer is: (c) \(0<\lambda<4\)

Key concepts

Understanding the Triangle Inequality Theorem

The Triangle Inequality Theorem is a fundamental principle in geometry. It states that in any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. This theorem guarantees that a triangle can actually exist with the given side lengths. If, for example, you had three sticks of lengths such that two of them added together are shorter than the third, you couldn't form a triangle.

In the context of IIT-JEE mathematics problems, this theorem plays a crucial role in defining the possible values of the sides of a triangle, thereby influencing the solution to the problems concerning geometric figures. A clear grasp of this theorem is a stepping stone to tackling more complex problems, like the one we're considering where inequalities are used alongside the variables representing the sides of the triangle.

Algebra in Triangles

Incorporating algebra into triangle problems opens a wide range of possibilities for analysis and solution. By representing the sides of a triangle with variables, we can apply algebraic manipulations to solve for unknowns or prove certain properties. As with the exercise in question, algebra allows us to express geometric constraints in terms of equations and inequalities.

Manipulating Expressions

As seen in the example problem, algebraic expressions involving the sides of a triangle can be manipulated to uncover relationships and solve for variables like \(\lambda\). By rewriting and rearranging terms, we gain insight into the possible values such variables can take. It's important for students to be comfortable rearranging and manipulating these algebraic equations as they form the basis for finding solutions in a range of geometry problems.

Understanding Equations and Inequalities

Sometimes, we encounter problems that do not have a unique solution, but rather a range of possible solutions. Understanding how to interpret and solve inequalities is key. The given problem illustrates how we can deduce a range for the variable \(\lambda\) through algebraic inequality manipulation, showing that it lies between two bounds. This kind of algebraic reasoning is invaluable for students aiming to excel in mathematics at an advanced level like the IIT-JEE.

Inequalities in Geometry

Inequalities are not just limited to algebra; they are also deeply intertwined with geometric concepts. In our exercise, inequalities are used to establish possible relationships between the lengths of the sides of a triangle. Through such relations, certain conclusions about the geometric properties and measurements can be drawn, which can lead in turn to the solving of the problem.

Setting Bounds

Geometric inequalities help in setting up bounds within which certain measurements must fall. For instance, the exercise under discussion does not provide a single, fixed value for \(\lambda\), but instead reveals that \(\lambda\) is within a certain interval. This kind of reasoning is not uncommon in IIT-JEE problems, where students may be asked to find the range of possible values rather than a specific number.

Implications for Figures

Understanding the implications of these inequalities on the geometric figures themselves is another vital skill. Inequalities can inform us about the nature of angles, sides, and other elements in the context of the shapes involved. Thus, a good grasp of inequalities in geometry is essential for students to successfully navigate through the more challenging problems of the IIT-JEE Mathematics section. Using inequalities in geometry, just like in algebra, requires strategic thinking and a methodical approach to determine which principles apply and how to best use them to reach a solution.