Question

If $a,b,c\in R$ and 1 is a root of equation $a{x}^2+bx+c=0$, then the curve $y=4a{x}^2+3bx+2c,a\ne 0$ 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 $a{x}^2+bx+c=0$
$\Rightarrow a{x}^2=-bx-c$
Now, consider
$y=4a{x}^2+3bx+2c$
$=4\left[-bx-c\right]+3bx+2c$
$=4bx-4c+3bx+2c$
$=-bx-2c$
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.