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Question
If a,b,c are in AP whereas x,y,z are in GP , what is the value of x(b−c).y(c−a).z(a−b)?
If a,b,c are in AP whereas x,y,z are in GP , what is the value of x(b−c).y(c−a).z(a−b)?
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Solution
The correct option is A 1
If a, b, c are in AP.
Hence, 2b = a + c
And x, y, z in GP.
Lets assume, common ratio is P, so y = xP and z=xP2
So,
x(b−c).y(c−a).z(a−b)=x(b−c).(xP)(c−a).(xP2)(a−b)=x(b−c).x(c−a).P(c−a).x(a−b).P(2a−2b)=x0.P(c+a−2b)=x0.P(2b−2b)=x0.P0=1
If a, b, c are in AP.
Hence, 2b = a + c
And x, y, z in GP.
Lets assume, common ratio is P, so y = xP and z=xP2
So,
x(b−c).y(c−a).z(a−b)=x(b−c).(xP)(c−a).(xP2)(a−b)=x(b−c).x(c−a).P(c−a).x(a−b).P(2a−2b)=x0.P(c+a−2b)=x0.P(2b−2b)=x0.P0=1
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