If the roots of the equation ax2+bx+c=0 are in the ratio m:n , then

Question

If the roots of the equation ax2+bx+c=0 are in the ratio m:n , then (A) mnb2=ac (m + n)2 (B) b2(m + n)=mn (C) m+n=b2mn (D) mnc2=ab (m + n)2

Tardigrade Admin · Accepted Answer

Let the roots are mr and nr mr+nr=(- b/a) ⇒ r(m + n)=(- b/a) ⇒ r2(m + n)2=(b2/a2) ldots (1) and mnr2=(c/a) ldots (2) Eliminating r from (1) &(2) , we get b2mn=ca(m + n)2

Q. If the roots of the equation $a x^{2} + b x + c = 0$ are in the ratio $m : n$ , then

$mn b^{2} = a c (m + n)^{2}$

$b^{2} (m + n) = mn$

$m + n = b^{2} mn$

$mn c^{2} = ab (m + n)^{2}$

Let the roots are $m r$ and $n r$
$m r + n r = \frac{- b}{a}$
$\Rightarrow r (m + n) = \frac{- b}{a}$
$\Rightarrow r^{2} (m + n)^{2} = \frac{b ^{2}}{a ^{2}} \dots (1)$
and $mn r^{2} = \frac{c}{a} \dots (2)$
Eliminating $r$ from $(1) & (2)$ , we get
$b^{2} mn = c a (m + n)^{2}$

Q. If the roots of the equation ax2+bx+c=0 are in the ratio m:n , then

mnb2=ac(m+n)2

b2(m+n)=mn

m+n=b2mn

mnc2=ab(m+n)2