Question:
Let a, b, c, d and p be any non zero distinct real numbers such that
\((a^2+b^2+c^2)p^2-2(ab+bc+cd)p+(b^2+c^2+d^2)=0\). Then :
Let a, b, c, d and p be any non zero distinct real numbers such that
\((a^2+b^2+c^2)p^2-2(ab+bc+cd)p+(b^2+c^2+d^2)=0\). Then :
\((a^2+b^2+c^2)p^2-2(ab+bc+cd)p+(b^2+c^2+d^2)=0\). Then :
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Write the left side as a sum of three squares: \((ap-b)^2+(bp-c)^2+(cp-d)^2=0\).
Updated On: Nov 5, 2026
- a, c, p are in G.P.
- a, b, c, d are in G.P.
- a, b, c, d are in A.P.
- a, c, p are in A.P.
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The Correct Option is B
Solution and Explanation
Expand \((ap-b)^2+(bp-c)^2+(cp-d)^2=(a^2+b^2+c^2)p^2-2(ab+bc+cd)p+(b^2+c^2+d^2)\).
So the given equation says the sum of three squares of real numbers is 0, hence each is 0:
\(ap=b,\ bp=c,\ cp=d\).
Therefore \(\dfrac ba=\dfrac cb=\dfrac dc=p\), i.e. a, b, c, d are in G.P. with common ratio p.
Answer: (2) a, b, c, d are in G.P.
So the given equation says the sum of three squares of real numbers is 0, hence each is 0:
\(ap=b,\ bp=c,\ cp=d\).
Therefore \(\dfrac ba=\dfrac cb=\dfrac dc=p\), i.e. a, b, c, d are in G.P. with common ratio p.
Answer: (2) a, b, c, d are in G.P.
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