Problem 88 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If the two equations \(a x^{2}+b x+c=0\) and \(2 x^{2}-3 x+\) \(4=0\) have a common root, then (A) \(6 a=4 b=-3 c\) (B) \(3 a=-4 b=3 c\) (C) \(6 a=-4 b=3 c\) (D) None of these

Step-by-Step Solution

Answer

Correct answer is (C) \(6a = -4b = 3c\).

Step 1: Understand common roots

A common root between two quadratic equations means that there is a value of \(x\) that satisfies both equations. For equations \(ax^2 + bx + c = 0\) and \(2x^2 - 3x + 4 = 0\) to have a common root, the discriminants of both must equal for that root.

Step 2: Assume a common root

Let the common root be \(\alpha\). Then \(a\alpha^2 + b\alpha + c = 0\) and \(2\alpha^2 - 3\alpha + 4 = 0\). Substituting in these equations can later help in deriving relationships between the coefficients.

Step 3: Use the result of common root from one equation

Since \(\alpha\) is a root of the second equation, it satisfies \(2\alpha^2 - 3\alpha + 4 = 0\). Rearrange to express \(\alpha\) in terms of its coefficients for substitution. However, the exact solution here will derive equations from known relations of coefficient equality due to roots.

Step 4: Express conditions for common roots

From quadratic theory, if two quadratics have a common root, any linear combination of the two also has a common root. If you subtract one equation from another, multiples of \(a,b,c\) should match each term of the second equation.

Step 5: Set proportional relationships

Let's assume \(l1\times( ax^2+bx+c) + l2\times(2x^2 - 3x + 4) = 0\). From this, by matching coefficients, we derive relationships such as \(l1a + 2l2 = 0\), \(l1b -3l2 = 0\), \(l1c + 4l2 = 0\). Solving each helps find proportionality.

Step 6: Solve for coefficients

From the relationships derived, solve for \(l1\) and \(l2\) to find that \(6a, -4b,3c\) must be proportional to match conditions of having the common root. Evaluate the options.

Step 7: Evaluate options

Given the relationships from Step 6, verify against the options: Check which of \(6a, -4b, 3c\) matches the derived conditions that suggest a common root. Verify option C.

Key Concepts

Common Roots

When solving quadratic equations, understanding the concept of common roots is very important. A common root implies that there is one specific value of \(x\) that satisfies both quadratic equations. It means, basically, that both equations share a solution.
Let's break it down further with the example of the equations \(ax^2 + bx + c = 0\) and \(2x^2 - 3x + 4 = 0\). To solve for a common root, we define this shared solution as \(\alpha\).
Now, substituting \(\alpha\) into both equations gives us two equations: \(a\alpha^2 + b\alpha + c = 0\) and \(2\alpha^2 - 3\alpha + 4 = 0\). Solving these two equations helps establish conditions under which a root can be common, playing an essential role in comparing coefficients in the following steps.

Coefficient Comparison

To determine if two quadratics have a common root, we often rely on comparing their coefficients after substituting the common root into each equation.
This comparison involves matching terms and applying the conditions of proportionality for each term. For instance, if \(\alpha\) is a common root of \(ax^2 + bx + c = 0\) and \(2x^2 - 3x + 4 = 0\), then:

• \(a\alpha^2 + b\alpha + c = 2\alpha^2 - 3\alpha + 4 = 0\)

From here, to find out how coefficients are interrelated, manipulate these equations. It will allow you to derive new relationships between \(a, b,\) and \(c\), making it easier to solve further for proportionality.

Proportional Relationships

Understanding proportional relationships is key when dealing with common roots in quadratic equations. It involves expressing coefficients in ratios that align between the two equations.
For our equations \(ax^2 + bx + c = 0\) and \(2x^2 - 3x + 4 = 0\), identify constants \(l_1\) and \(l_2\) such that their linear combination eliminates the common root. This means setting:

• \(l1(a\alpha^2 + b\alpha + c) + l2(2\alpha^2 - 3\alpha + 4) = 0\)

Next, we derive these coefficient conditions through equations, like:

• \(l1a + 2l2 = 0\)
• \(l1b -3l2 = 0\)
• \(l1c + 4l2 = 0\)

Solving these equations allows for establishing proportional constants such as option C: \(6a = -4b = 3c\). These steps illustrate how proportionality aids in revealing the true relationship between coefficients when finding common roots.