5. If (a + b)(a + d) = (ac + bc)2, then prove that a: c:: b: d.
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Answer:
Given a,b,c,d are in continued proportion
⟹
b
a
=
c
b
=
d
c
=k(say)
⟹c=dk,b=ck=k
2
d,a=bk=k
3
d
LHS=(a+d)(b+c)−(a+c)(b+d)=(k
3
d+d)(k
2
d+kd)−(k
3
d+kd)(k
2
d+d)
=d
2
(k
5
+k
2
+k
4
+k−k
5
−2k
3
−k)
=k
2
d
2
(k−1)
2
=(k
2
d−kd)
2
=(b−c)
2
=RHS
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