If-p-q-r-0-and-A-p-2-q-2-r-2-2-pq-2-qr-2-rp-2-B-q-2-pr-p-2-q-2-r-2-Find-A-2-4B-

Question Number 207664 by efronzo1 last updated on 22/May/24

If p + q + r = 0 and

A = \frac{{(p^{2} + q^{2} + r^{2})}^{2}}{{(pq)}^{2} + {(qr)}^{2} + {(rp)}^{2}}

B = \frac{q^{2} - pr}{p^{2} + q^{2} + r^{2}} .

Find A^{2} - 4 B

Answered by A5T last updated on 22/May/24

{(p + q + r)}^{2} = p^{2} + q^{2} + r^{2} + 2 (p q + q r + r p)

\Rightarrow p^{2} + q^{2} + r^{2} = - 2 (p q + q r + r p)

{(p q)}^{2} + {(q r)}^{2} + {(r p)}^{2} = {(p q + q r + r p)}^{2} - 2 p q r (p + q + r)

\Rightarrow A^{2} = \frac{16 {(p q + q r + r p)}^{4}}{{(p q + q r + r p)}^{4}} = 16

q^{2} = {(p + r)}^{2} = p^{2} + r^{2} + 2 p r \Rightarrow q^{2} - p r = p^{2} + r^{2} + p r

p^{2} + q^{2} + r^{2} = 2 (p^{2} + r^{2} + p r)

\Rightarrow B = \frac{q^{2} - p r}{p^{2} + q^{2} + r^{2}} = \frac{1}{2}

\Rightarrow A - 4 B = 16 - 4 (\frac{1}{2}) = 14

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