Question

If $a, b, c \in R$ and 1 is a root of equation $ax^2 + bx + c = 0$, then the curve $y = 4ax^2 + 3 bx + 2c, a \ne 0$ intersect x-axis at

• A. two distinct points whose coordinates are always rational numbers
• B. no point
• C. exactly two distinct points
• D. exactly one point

Answer 1

Answer: (D)

Given $ax^2 + bx + c = 0$
$\Rightarrow ax^{2} = -bx - c$
Now, consider
$y = 4ax^{2} + 3bx + 2c$
$= 4 \left[-bx - c\right] +3 bx + 2c$
$= 4bx - 4c + 3bx + 2c$
$= -bx-2c$
Since, this curve intersects x-axis
$\therefore $ put $y = 0$, we get
$-bx - 2c = 0 \Rightarrow - bx = 2c$
$\Rightarrow x = \frac{-2c}{b}$
Thus, given curve intersects x-axis at exactly one point.