Problem 128 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(k\) be any point such that \(k \in R\) and \(\alpha, \beta\) are the roots of the quadratic equation \(f(x)=a x^{2}+b x+c=0\). If \(k\) lies outside and is less than both the roots then the equation must have real and distinct roots and the sign of \(f(k)\) is same as the sign of ' \(a\) '. Also, \(k\) is less than the \(x\)-coordinate of the vertex of the parabola \(y=a x^{2}+b x+c\). If \(k\) lies between both the roots, then the sign of \(f(k)\) is opposite to the sign of ' \(a\) '. If \(k\) lies outside and is greater than both the roots, then the sign of \(f(k)\) is same as the sign of ' \(a\) '. Also, \(k\) is greater than the \(x\)-coordinate of the vertex of the parabola \(y=a x^{2}+b x+c\). If both the roots of the equation lie between two real numbers \(k_{1}\) and \(k_{2}\), then equation must have real and distinct roots and the sign of \(f\left(k_{1}\right)\) and \(f\left(k_{2}\right)\) is same as the sign of \(a\). Also, the \(x\)-coordinate of the vertex of the parabola \(y=a x^{2}+b x+c\) lies between \(k_{1}\) and \(k_{2}\). The values of ' \(a\) ' so that 6 lies between the roots of the equation \(x^{2}+2(a-3) x+9=0\), are (A) \(a>-\frac{3}{4}\) (B) \(a<-\frac{3}{4}\) (C) \(a>\frac{3}{4}\) (D) \(a<\frac{3}{4}\)

Step-by-Step Solution

Answer

The correct answer is (B) \(a < -\frac{3}{4}\).

Step 1: Find the Vertex

The quadratic equation is given by \(x^2 + 2(a-3)x + 9 = 0\). For a quadratic equation \(ax^2 + bx + c = 0\), the x-coordinate of the vertex is given by \(\frac{-b}{2a}\). Here, \(a = 1\) and \(b = 2(a-3)\); hence, the x-coordinate is \(\frac{-2(a-3)}{2}\) which simplifies to \(3-a\).

Step 2: Condition for 6 Between Roots

If 6 is between the roots, then \(f(6) < 0\). We substitute \(x = 6\) into the quadratic: \(f(6) = 36 + 12(a-3) + 9\). Simplifying gives \(f(6) = 45 + 12a - 36\), which simplifies further to \(9 + 12a\). For \(f(6) < 0\), \(9 + 12a < 0\).

Step 3: Solve Inequality

Solving \(9 + 12a < 0\), we subtract 9 from both sides: \(12a < -9\). Now, divide by 12: \(a < -\frac{3}{4}\).

Step 4: Conclusion

The inequality \(a < -\frac{3}{4}\) means that when \(a\) is less than \(-\frac{3}{4}\), 6 lies between the roots of the quadratic.

Key Concepts

Real and Distinct Roots

When solving a quadratic equation of the form \(ax^2 + bx + c = 0\), one of the key properties we often look for is whether the roots are real and distinct. Real and distinct roots occur when the discriminant \(b^2 - 4ac\) is greater than zero. This indicates that the quadratic equation will intersect the \(x\)-axis at two different points, which are the roots or solutions of the equation.

Here are some situations related to real and distinct roots:

• The quadratic has two distinct, real roots when \(b^2 - 4ac > 0\).
• Both roots are equal (repeated root) when \(b^2 - 4ac = 0\).
• There are no real roots (imaginary roots) when \(b^2 - 4ac < 0\).

Understanding these scenarios helps in graphing the quadratic function and predicting the nature of its graph, which shows two intersections with the \(x\)-axis.

Vertex of a Parabola

The vertex of a parabola given by \(ax^2 + bx + c\) is a crucial concept in quadratic equations. The vertex represents the highest or lowest point on the parabola, depending on whether it opens upwards or downwards. It's a point of symmetry, providing insights into the shape and direction of the graph.

To find the vertex, you use the formula for the \(x\)-coordinate of the vertex: \(x = \frac{-b}{2a}\). For example, in the given equation \(x^2 + 2(a-3)x + 9 = 0\), our \(a = 1\) and \(b = 2(a-3)\). Thus, the \(x\)-coordinate of the vertex is \(3 - a\).

The vertex tells us:

• The line of symmetry of the parabola, which is vertical and goes through \(x = \frac{-b}{2a}\).
• Whether the parabola has a minimum (if \(a > 0\)) or a maximum (if \(a < 0\)).
• Where an extremum of the parabola sits on the \(x\)-axis, aiding in identifying the range of the function.

Sign of Quadratic Function

The sign of a quadratic function refers to whether the output \(f(x)\) is positive or negative for given values of \(x\). This feature is particularly relevant when discussing regions outside or between the roots of a quadratic.

Let's delve into the effect of the constant \(a\) on the function's sign:

• If \(k < \alpha\) and \(k < \beta\) (both roots), or \(k > \alpha\) and \(k > \beta\), then the sign of \(f(k)\) tends to match the sign of \(a\).
• When \(k\) lies between \(\alpha\) and \(\beta\), the sign of \(f(k)\) is usually opposite to \(a\).

These insights are crucial, especially for determining intervals where the quadratic is above or below the \(x\)-axis, which can be used in solving inequalities and optimization problems. By analyzing where the function changes sign, one can understand the regions of increase and decrease, as well as the overall behavior of the parabola.