if a,b,c and d are in proportion prove that a-b/c-d= root of 3a²+8b²/3c²+8d²
Answers
Answer:
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Step-by-step explanation:
Given a,b,c,d are in continued proportion
⟹ba=cb=dc=k(say)
⟹c=dk,b=ck=k2d,a=bk=k3d
RHS=(b−ca−b)3=(k2d−kdk3d−k2d)3=k3
LHS=da=k3
LHS=RHS
Hence Proved
Given : if a,b,c and d are in proportion
To Find : prove that a-b/c-d=√ (3a²+8b²/3c²+8d²)
Solution:
a,b,c and d are in proportion
a/b = b/c = c/d = k
=> a = bk ,
c = dk
a-b/c-d=√ (3a²+8b²/3c²+8d²)
LHS = a-b/c-d
= (bk - b) /(dk - d)
= b(k - 1)/d(k - 1)
= b/d
RHS = √ (3a²+8b²/3c²+8d²)
= √ (3(bk)²+8b²/3(dk)²+8d²)
= √ (3b²k²+8b²/3 d²k²+8d²)
= √b²(3k² + 8)/d²(3k² + 8)
= √b²/d²
= b/d
LHS = RHS = b/d
QED
Hence Proved
a-b/c-d=√ (3a²+8b²/3c²+8d²)
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