If z1 and z2 are the two complex roots of equal magnitude and their arguments differ by (π/2), of the quadratic equation a x2+b x+c=0(a ≠ 0) then a (in terms of b and c ) is

Question

If z1 and z2 are the two complex roots of equal magnitude and their arguments differ by (π/2), of the quadratic equation a x2+b x+c=0(a ≠ 0) then a (in terms of b and c ) is (A) (b2/2 c) (B) (b2/c) (C) (b/2 c) (D) None of these

Tardigrade Admin · Accepted Answer

z1+z2=-(b/a) (1) z1 z2=(c/a) (2) z2=i z1 (3) From Eq. (1) and (2) z1(1+i)=(-b/a) ⇒ z1=(-b/2 a)(1-i) ⇒ z12=(b2/4 a2)(-2 i)=(-b2/2 a2) i From Eq. (2) and (3) z12=(c/a i)=(-c/a) i ⇒ (-b2/2 a2) i=(c/a) i ⇒ a=(b2/2 c)

Q. If $z_{1}$ and $z_{2}$ are the two complex roots of equal magnitude and their arguments differ by $\frac{π}{2}$ , of the quadratic equation $a x^{2} + b x + c = 0 (a \neq = 0)$ then $a$ (in terms of $b$ and $c$ ) is

$\frac{b ^{2}}{2 c}$

$\frac{b ^{2}}{c}$

$\frac{b}{2 c}$

None of these

Q. If z1​ and z2​ are the two complex roots of equal magnitude and their arguments differ by 2π​, of the quadratic equation ax2+bx+c=0(a=0) then a (in terms of b and c ) is

2cb2​

cb2​

2cb​

None of these

z1​+z2​=−ab​(1) z1​z2​=ac​(2) z2​=iz1​(3) From Eq. (1) and (2) z1​(1+i)=a−b​ ⇒z1​=2a−b​(1−i) ⇒z12​=4a2b2​(−2i)=2a2−b2​i From Eq. (2) and (3) z12​=aic​=a−c​i ⇒2a2−b2​i=ac​i ⇒a=2cb2​

Q. If $z_{1}$ and $z_{2}$ are the two complex roots of equal magnitude and their arguments differ by $\frac{π}{2}$ , of the quadratic equation $a x^{2} + b x + c = 0 (a \neq = 0)$ then $a$ (in terms of $b$ and $c$ ) is

$\frac{b ^{2}}{2 c}$

$\frac{b ^{2}}{c}$

$\frac{b}{2 c}$