First slide

Theory of expressions

Question

If ax² + bx + c = 0 has imaginary roots and a - b + c > 0, then the set of points (x, y) satisfying the equation $|a (x^{2} + \frac{y}{a}) + (b + 1) x + c| = |{ax}^{2} + bx + c| + | x + y |$ consists of the region in the xy-plane which is

Moderate

A

on or above the bisector of I and III quadrant
B

on or above the bisector of II and IV quadrant
C

on or below the bisector of I and III quadrant
D

on or below the bisector of II and IV quadrant

Solution

$|a (x^{2} + \frac{y}{a}) + (b + 1) x + c| = |{ax}^{2} + bx + c| + | x + y |$
$\Rightarrow |({ax}^{2} + bx + c) + (x + y)| = |{ax}^{2} + bx + c| + | x + y |$ (1)
Now $f (x) = {ax}^{2} + bx + c = 0$ has imaginary roots and $a - b + c > 0 or f (- 1) > 0$
$\Rightarrow f (x) = {ax}^{2} + bx + c > 0$ for all real values of x
$\Rightarrow x + y \geq 0$
$\Rightarrow (x, y)$ lies on or above the bisector of II and IV quadrants

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