First slide

Cartesian plane

Question

The points $P (a, b + c), Q (b, c + a)$ and $R (c, a + b)$ are such that $P Q = Q R$ if $(a \neq b \neq c)$

Easy

A

a, b, c are in A.P
B

a, b, c are in G.P
C

a, b, c are in H.P
D

a, b, c are in A.G.P

Solution

$P Q = Q R \Rightarrow {(b - a)}^{2} + {(a - b)}^{2} = {(c - b)}^{2} + {(b - c)}^{2}$
$\Rightarrow {(a - b)}^{2} = {(b - c)}^{2} \Rightarrow a - b = b - c o r a - b = c - b$

i.e $= a + c = 2 b o r a = c$

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