If the cubic equation x3+p x2+q x+r=0 where p, q, r ∈ R has root a2, b2, c2 satisfying a2+b2=c2, then the value of (p3+8 r/p q) is equal to λ. Find the value of λ5.

Question

If the cubic equation x3+p x2+q x+r=0 where p, q, r ∈ R has root a2, b2, c2 satisfying a2+b2=c2, then the value of (p3+8 r/p q) is equal to λ. Find the value of λ5. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/9fd0c1309c797c42afaca107c919f7ff-.png" /> a2+b2+c2=-p ...(1) a2 b2+b2 c2+c2 a2=q ....(2) a2 b2 c2=-r ...(3) also given a2+b2=c2 ...(4) (1) and (4) ⇒ c2=-(p/2) put x =-( p /2) in the cubic to be t the answer

Q. If the cubic equation $x^{3} + p x^{2} + q x + r = 0$ where $p, q, r \in R$ has root $a^{2}, b^{2}, c^{2}$ satisfying $a^{2} + b^{2} = c^{2}$ , then the value of $\frac{p ^{3} + 8 r}{pq}$ is equal to $λ$ . Find the value of $λ^{5}$ .

$a^{2} + b^{2} + c^{2} = - p$ ...(1)
$a^{2} b^{2} + b^{2} c^{2} + c^{2} a^{2} = q$ ....(2)
$a^{2} b^{2} c^{2} = - r$ ...(3)
also given $a^{2} + b^{2} = c^{2}$ ...(4)
(1) and (4) $\Rightarrow c^{2} = - \frac{p}{2}$
put $x = - \frac{p}{2}$ in the cubic to be $t$ the answer

Q. If the cubic equation x3+px2+qx+r=0 where p,q,r∈R has root a2,b2,c2 satisfying a2+b2=c2, then the value of pqp3+8r​ is equal to λ. Find the value of λ5.

a2+b2+c2=−p ...(1) a2b2+b2c2+c2a2=q....(2) a2b2c2=−r ...(3) also given a2+b2=c2 ...(4) (1) and (4) ⇒c2=−2p​ put x=−2p​ in the cubic to be t the answer

Q. If the cubic equation $x^{3} + p x^{2} + q x + r = 0$ where $p, q, r \in R$ has root $a^{2}, b^{2}, c^{2}$ satisfying $a^{2} + b^{2} = c^{2}$ , then the value of $\frac{p ^{3} + 8 r}{pq}$ is equal to $λ$ . Find the value of $λ^{5}$ .

$a^{2} + b^{2} + c^{2} = - p$ ...(1)
$a^{2} b^{2} + b^{2} c^{2} + c^{2} a^{2} = q$ ....(2)
$a^{2} b^{2} c^{2} = - r$ ...(3)
also given $a^{2} + b^{2} = c^{2}$ ...(4)
(1) and (4) $\Rightarrow c^{2} = - \frac{p}{2}$
put $x = - \frac{p}{2}$ in the cubic to be $t$ the answer