Let p and q be two real numbers such that p + q = 3 and p4 + q4 = 369. Then ${\left( {{1 \over p} + {1 \over q}} \right)^{ - 2}}$ is equal to _________.
Answer (integer)
4
Solution
<p>$\because$ $p + q = 3$ ...... (i)</p>
<p>and ${p^4} + {q^4} = 369$ ...... (ii)</p>
<p>${\{ {(p + q)^2} - 2pq\} ^2} - 2{p^2}{q^2} = 369$</p>
<p>or ${(9 - 2pq)^2} - 2{(pq)^2} = 369$</p>
<p>or ${(pq)^2} - 18pq - 144 = 0$</p>
<p>$\therefore$ $pq = - 6$ or 24</p>
<p>But $pq = 24$ is not possible</p>
<p>$\therefore$ $pq = - 6$</p>
<p>Hence, $${\left( {{1 \over p} + {1 \over q}} \right)^{ - 2}} = {\left( {{{pq} \over {p + q}}} \right)^2} = {( - 2)^2} = 4$$</p>
About this question
Subject: Mathematics · Chapter: Complex Numbers and Quadratic Equations · Topic: Complex Numbers and Argand Plane
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