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). 

Answer 1

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