Problem 66 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(a<0\), the positive root of the equation \(x^{2}-2 a\) \(|x-a|-3 a^{2}=0\) is (A) \(a(-1-\sqrt{6})\) (B) \(a(-1+\sqrt{6})\) (C) \(a(1-\sqrt{2})\) (D) None of these

Step-by-Step Solution

Answer

The positive root is \(a(1-\sqrt{2})\), which corresponds to option (C).

Step 1: Analyze the Equation

The given equation is \(x^{2} - 2a|x-a| - 3a^{2} = 0\). We are asked to find the positive root for \(a < 0\). We need to consider different cases for the absolute value expression \(|x-a|\).

Step 2: Consider Case When \(x \geq a\)

If \(x \geq a\), then \(|x-a| = x-a\). Substitute into the equation to get \(x^2 - 2a(x-a) - 3a^2 = 0\). This simplifies to \(x^2 - 2ax + 2a^2 - 3a^2 = 0\), which simplifies further to \(x^2 - 2ax - a^2 = 0\).

Step 3: Solve the Quadratic Equation

The quadratic equation \(x^2 - 2ax - a^2 = 0\) can be solved using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(b = -2a\) and \(c = -a^2\). Calculate \(x = \frac{2a \pm \sqrt{(-2a)^2 - 4 \cdot 1 \cdot (-a^2)}}{2}\).

Step 4: Simplify the Root Expression

Calculate \((-2a)^2 = 4a^2\) and \(-4(-a^2) = 4a^2\). Thus, the discriminant is \(4a^2 + 4a^2 = 8a^2\). The possible roots are \(x = \frac{2a \pm 2\sqrt{2}a}{2}\). Simplifying gives \(x = a(1 \pm \sqrt{2})\).

Step 5: Determine the Positive Root

Since \(a < 0\), the expression \(a(1 - \sqrt{2})\) will be positive because \(1-\sqrt{2}\) is negative (approximately \(-0.41\)), making the product positive.

Key Concepts

Absolute Value

The absolute value of a number indicates its distance from zero on the number line, without considering the direction. For any real number \(x\), the absolute value is denoted by \(|x|\). It is defined as:

• \(|x| = x\) if \(x \geq 0\)
• \(|x| = -x\) if \(x < 0\)

This piecewise definition is crucial when dealing with expressions like \(|x-a|\), as in our quadratic equation.

In the exercise, we encountered the expression \(|x-a|\), which can have two forms depending on whether \(x\) is greater than or less than \(a\). This translates to considering different potential cases for our problem:

• If \(x \geq a\), then \(|x-a| = x-a\).
• If \(x < a\), then \(|x-a| = a-x\).

Only by analyzing both situations can we correctly solve the equation.

Discriminant

The discriminant of a quadratic equation helps in determining the nature of its roots. For a standard quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant, denoted as \(D\), is given by the formula:

• \(D = b^2 - 4ac\)

The value of the discriminant indicates:

• If \(D > 0\), there are two distinct real roots.
• If \(D = 0\), there is exactly one real root.
• If \(D < 0\), there are no real roots, but two complex roots.

In our quadratic equation step, when simplifying \((x^2 - 2ax - a^2 = 0)\), we calculated the discriminant to be \(8a^2\), which is always non-negative for any real value of \(a\).

This positive discriminant ensures that the equation has two distinct real roots. It is crucial for finding the positive root of our equation, especially since \(a < 0\).

Quadratic Formula

The quadratic formula is a fundamental tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). The formula is:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula provides the solutions (roots) of the quadratic equation by calculating based on the coefficients \(a\), \(b\), and \(c\).

In our problem, after discerning the correct absolute value consideration, the quadratic equation became \(x^2 - 2ax - a^2 = 0\). Applying the quadratic formula to solve it, we set:

• \(a = 1\)
• \(b = -2a\)
• \(c = -a^2\)

We found the roots to be \(x = a(1 \pm \sqrt{2})\). Since \(a < 0\), it allows us to deduce that \(x = a(1 - \sqrt{2})\) is the positive root due to multiplication rules with negative numbers. This emphasizes the importance of analyzing all components within the quadratic formula accurately to arrive at the correct solution.