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Question
Find the fourth proportion to the following:
(p2q - qr2 ), (pqr - pr2 ) and (pq2 - pr2)
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Solution
Let x be the fourth proportion
`("p"^2"q" - "qr"^2) : ("pqr" - "pr"^2) :: ("pq"^2 - "pr"^2) : x`
⇒ `("p"^2"q" - "qr"^2) xx "x" = ("pqr" - "pr"^2) xx ("pq"^2 - "pr"^2)`
⇒ x = `(("pqr" - "pr"^2) xx ("pq"^2 - "pr"^2))/(("p"^2"q" - "qr"^2))`
⇒ x = `("pr"("q" - "r") xx "p"("q"^2 - "r"^2))/("q"("p"^2 - "r"^2))`
⇒ x = `("pr"("q - r") xx "p"("q - r")("q + r"))/("q"("p"^2 - "r"^2))`
⇒ x = `("p"^2"r" ("q" - "r")^2 ("q" + "r"))/("q" ("p"^2 - "r"^2))`
The fourth proportion is `("p"^2"r" ("q" - "r")^2 ("q" + "r"))/("q" ("p"^2 - "r"^2))`
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