If a, b and c are the roots of equation x3 - px2 + qx - r = 0 , then the value of (1/a2) + (1/b2) + (1/c2) is

Question

If a, b and c are the roots of equation x3 - px2 + qx - r = 0 , then the value of (1/a2) + (1/b2) + (1/c2) is (A)  (p2-2qr/r2)  (B)  (q2-2pr/r2)  (C)  (r2-2pq/q2)  (D)  (r2-2pq/p2)

Tardigrade Admin · Accepted Answer

Since, a, b and c are the roots of equation x3-p x2+q x-r=0 ∴ a+b+c=p, a b+b c+c a=q, a b c=r Now, (1/a2)+(1/b2)+(1/c2)=(a2 b2+b2 c2+c2 a2/(a b c)2) =((a b)2+(b c)2+(c a)2/(a b c)2) =((a b+b c+c a)2-2(a b2 c+b c2 a+c a2 b)/(a b c)2) =((a b+b c+c a)2-2 a b c(a+b+c)/(a b c)2) =(q2-2 r p/r2) =(q2-2 p r/r2)

Q. If $a, b$ and $c$ are the roots of equation $x^{3} - p x^{2} + q x - r = 0$ , then the value of $\frac{1}{a ^{2}} + \frac{1}{b ^{2}} + \frac{1}{c ^{2}}$ is

$\frac{p ^{2} - 2 q r}{r ^{2}}$

$\frac{q ^{2} - 2 p r}{r ^{2}}$

$\frac{r ^{2} - 2 pq}{q ^{2}}$

$\frac{r ^{2} - 2 pq}{p ^{2}}$

Q. If a,b and c are the roots of equation x3−px2+qx−r=0 , then the value of a21​+b21​+c21​ is

r2p2−2qr​

r2q2−2pr​

q2r2−2pq​

p2r2−2pq​

Since, a,b and c are the roots of equation x3−px2+qx−r=0 ∴a+b+c=p,ab+bc+ca=q,abc=r Now, a21​+b21​+c21​=(abc)2a2b2+b2c2+c2a2​ =(abc)2(ab)2+(bc)2+(ca)2​ =(abc)2(ab+bc+ca)2−2(ab2c+bc2a+ca2b)​ =(abc)2(ab+bc+ca)2−2abc(a+b+c)​ =r2q2−2rp​ =r2q2−2pr​

Q. If $a, b$ and $c$ are the roots of equation $x^{3} - p x^{2} + q x - r = 0$ , then the value of $\frac{1}{a ^{2}} + \frac{1}{b ^{2}} + \frac{1}{c ^{2}}$ is

$\frac{p ^{2} - 2 q r}{r ^{2}}$

$\frac{q ^{2} - 2 p r}{r ^{2}}$

$\frac{r ^{2} - 2 pq}{q ^{2}}$

$\frac{r ^{2} - 2 pq}{p ^{2}}$