Problem 77 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(a, b, c \in \mathrm{R}\) and 1 is a root of equation \(a x^{2}+b x+c=0\) then the curve \(y=4 a x^{2}+3 b x+2 c, a \neq 0\) intersect \(x\)-axis at [Online May 26, 2012] (a) two distinct points whose coordinates are always rational numbers (b) no point (c) exactly two distinct points (d) exactly one point

Step-by-Step Solution

Answer

(c) exactly two distinct points

Step 1: Analyze the Root and Equation

Since 1 is a root of the equation \(a x^{2} + b x + c = 0\), this implies that if we substitute \(x = 1\) into the equation, it should equal zero. Therefore, \(a(1)^2 + b(1) + c = a + b + c = 0\). This will be an important equation to use later.

Step 2: Set Up the Intersection with the X-axis

To find where the curve \(y = 4ax^2 + 3bx + 2c\) intersects the x-axis, we need to set \(y = 0\). Thus, the equation becomes \(4ax^2 + 3bx + 2c = 0\).

Step 3: Substitute Derived Value into the Equation

Using \(a + b + c = 0\), we can express \(c\) as \(c = -a - b\). Substitute this into the equation obtained in Step 2: \(4ax^2 + 3bx + 2(-a-b) = 0\), which simplifies to \(4ax^2 + 3bx - 2a - 2b = 0\).

Step 4: Simplify and Solve the Quadratic Equation

The new equation is \(4ax^2 + 3bx - 2a - 2b = 0\). We need to find the discriminant \(D = B^2 - 4AC\): - Here, \(A = 4a\), \(B = 3b\), and \(C = -2a - 2b\).- Calculate \(D = (3b)^2 - 4(4a)(-2a - 2b)\).- Simplifying, \(D = 9b^2 + 32a^2 + 32ab\).

Step 5: Determine the Nature of the Roots

Since the discriminant \(D = 9b^2 + 32a(a + b)\), and substituting \(a+b = -c\), the only requirement for having real roots is that \(D > 0\). - Since \(D\) is a sum of squares and positive terms, it is always greater than or equal to 0.- Therefore, the curve intersects the x-axis at two distinct points, unless if both terms are zero, simplifying to exactly one point.

Key Concepts

Root of Equation

A root of a quadratic equation is simply the value of the variable that, when substituted into the equation, makes the equation true. In other words, it's the value of \(x\) that makes \(ax^2 + bx + c = 0\) equal to zero.
For example, if you have \(1\) as a root of the equation \(ax^2 + bx + c = 0\), this means:

• When \(x = 1\) is substituted into the equation, the result will be zero.
• This gives us the equation: \(a(1)^2 + b(1) + c = a + b + c = 0\).

Hence, a root is essential for resolving the quadratic equation as it provides critical insights into the behavior of the equation.

Discriminant

The discriminant is a powerful tool to determine the nature of the roots of a quadratic equation. It's given by the formula \(D = B^2 - 4AC\). The discriminant helps to decide whether the roots are real or imaginary.
For the quadratic equation \(Ax^2 + Bx + C = 0\), let's see what the discriminant tells us about its roots:

• If \(D > 0\), the equation has two distinct real roots. This means the curve will intersect the x-axis at two points.
• If \(D = 0\), there is exactly one real root (also called a repeated root). The curve touches the x-axis at just one point.
• If \(D < 0\), no real roots exist, implying the curve does not intersect the x-axis.

In our solution, the discriminant \(D = 9b^2 + 32a^2 + 32ab\), which is always greater than zero due to it being a sum of positive terms, leading us to expect two distinct intersection points on the x-axis.

Intersection with X-axis

The intersection of a curve with the x-axis is determined by finding where \(y = 0\). This is because at the x-axis, the \(y\) value is always zero. For a given quadratic equation like \(y = 4ax^2 + 3bx + 2c\), setting \(y = 0\) allows us to solve for \(x\), the points where the curve intersects the x-axis.
Here's how you normally proceed:

• Substitute \(y = 0\) into the equation to find the values of \(x\).
• The equation becomes \(4ax^2 + 3bx + 2c = 0\).
• Depending on the nature of the discriminant, identify whether the curve intersects the x-axis at two points, one point, or not at all.

In our example, after simplifying with the condition \(a + b + c = 0\), the quadratic equation becomes \(4ax^2 + 3bx - 2a - 2b = 0\), which intersects the x-axis at two distinct points, given that we've identified a positive discriminant.