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Question
Given four quantities p, q, r and s are in proportion, show that
q2(p - r) : rs (q - s) =(p2- q2- pq): ( r2-s2-rs).
Solution
p, q, r and s are 1n proportion
then, p : q :: r : s
Let `"p"/"q" = "r"/"s" = "k"`
Then p = kq and r =ks
Now, we have to prove that
`(("p" - "r")"q"^2)/(("q - s")"rs") = ("p"^2 - "q"^2 - "pq")/("r"^2 - "s"^2 - "rs")`
LHS
`= (("p" - "r")"q"^2)/(("q - s")"rs")`
`= (("kq" - "ks")"q"^2)/(("q - s")"ks" xx "s")`
`= ("k"("q - s")"q"^2)/("ks"^2 ("q - s"))`
`= "q"^2/"s"^2`
RHS
`= ("p"^2 - "q"^2 - "pq")/("r"^2 - "s"^2 - "rs")`
`= ("k"^2"q"^2 - "q"^2 - "kq" xx "q")/("k"^2"s"^2 - "s"^2 - "ks" xx "s")`
`= ("q"^2("k"^2 - 1 - "k"))/("s"^2("k"^2 - 1 - "k"))`
`= "q"^2/"s"^2`
LHS = RHS
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