Problem 54 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(x^{2}-(a+b+c) x+(a b+b c+c a)=0\) has imaginary roots, where \(a, b, c \in R^{+}\), then \(\sqrt{a}, \sqrt{b}, \sqrt{c}\) (A) can be the sides of a triangle (B) cannot be the sides of a triangle (C) nothing can be said (D) None of these

Step-by-Step Solution

Answer

Option (B): Cannot be the sides of a triangle.

Step 1: Examine the condition for imaginary roots

A quadratic equation has imaginary roots if its discriminant \(D\) is less than zero. The discriminant for the equation \(x^2 - (a+b+c)x + (ab+bc+ca) = 0\) is given by \((a+b+c)^2 - 4(ab+bc+ca)\). We need this discriminant to be less than zero for the roots to be imaginary.

Step 2: Simplify the discriminant expression

Let's expand \((a+b+c)^2\) as \(a^2 + b^2 + c^2 + 2ab + 2bc + 2ca\). Subtract \(4(ab+bc+ca)\) from this, which yields the expression: \(a^2 + b^2 + c^2 + 2ab + 2bc + 2ca - 4ab - 4bc - 4ca = a^2 + b^2 + c^2 - 2ab - 2bc - 2ca\). This can be further simplified to \((a-b)^2 + (b-c)^2 + (c-a)^2\) which needs to be less than zero.

Step 3: Apply conditions for a positive sum

Since \((a-b)^2 + (b-c)^2 + (c-a)^2\) involves squares, each term is non-negative. Therefore, for the entire sum \((a-b)^2 + (b-c)^2 + (c-a)^2\) to be less than zero, it is impossible unless all terms are zero, which implies \(a = b = c\).

Step 4: Analyze the implications of equal terms

If \(a = b = c\), then \((a-b)^2 = (b-c)^2 = (c-a)^2 = 0\), and \((a,b,c)\) are not different, which implies that the squared parameters \((\sqrt{a}, \sqrt{b}, \sqrt{c})\) represent identical values that can indeed form a degenerate triangle (as they are equivalent sides of zero length). Thus, equality or mathematical constraints force the inequalities necessary for triangle inequality theorem failure. \(\sqrt{a}, \sqrt{b}, \sqrt{c}\) cannot be the sides of a non-degenerate triangle.

Key Concepts

Imaginary Roots

In the realm of complex numbers, imaginary roots are an interesting concept that often gets students scratching their heads. When we consider a quadratic equation, such as \[ x^2 - (a+b+c)x + (ab+bc+ca) = 0 \], it will have imaginary roots under a specific condition: when its discriminant \(D\) is less than zero. The discriminant is calculated as \[ D = (a+b+c)^2 - 4(ab+bc+ca) \]. Imaginary roots occur because the square root of a negative number is not defined in the real numbers, moving us into the complex numbers space, where these roots are evident.

• Imaginary roots indicate the quadratic does not intersect the x-axis.
• They are complex conjugates, with real parts being equal and imaginary parts having opposite signs.

Understanding this helps in identifying scenarios where the given quadratic can't be resolved into real number solutions. Imaginary roots are a natural occurrence in mathematics, enriching our understanding of equations and their graphical representations.

Quadratic Equations

A quadratic equation enjoys a special place in mathematics. It is of the form \[ ax^2 + bx + c = 0 \], where 'a', 'b', and 'c' are constants and \( x \) represents an unknown variable.Quadratic equations are solved using methods like:

• Factoring: Breaking down into simpler binomials.
• Completing the square: Reordering terms to form a perfect square binomial.
• Quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), a straightforward approach for any standard quadratic.

Discovery of Solutions: Real vs Imaginary
The discriminant of a quadratic equation determines the nature of its roots. For the equation \( ax^2 + bx + c = 0 \), if the discriminant \( b^2 - 4ac \) is:

• Positive, we have two distinct real roots.
• Zero, we have exactly one real root (or a repeated root).
• Negative, indicating two complex conjugate (imaginary) roots.

Quadratic equations pop up in a multitude of scenarios, from physics to finance, offering a method to model different real-world problems.

Triangle Inequality

The triangle inequality is a fundamental theorem in geometry. It states that for any triangle with sides of lengths \( a \), \( b \), and \( c \), the sum of the lengths of any two sides must be greater than the length of the remaining side.This can be represented as:

• \( a + b > c \)
• \( b + c > a \)
• \( c + a > b \)

If these conditions are not met, a triangle can't be formed. However, these rules are not strictly met when dealing with degenerate triangles, where the sides are such that they just touch each other without forming an enclosed area.Application in Problem Scenarios
In the context of roots derived from quadratic equations, like \( \sqrt{a}, \sqrt{b}, \sqrt{c} \), the triangle inequality helps determine whether these computed values can serve as actual side lengths.When we find that they cannot fulfill the inequality conditions (as in the case of equating them like \( a = b = c \)), it becomes clear they can't form a non-degenerate triangle. Instead, they suggest a degenerate scenario or do not align as logical triangle sides. Thus, the triangle inequality acts as a validator for possible triangle formation scenarios.