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Question
If a, b, c are in A.P. and x, y, z are in G.P., then the value of xb − c yc − a za − b is
Options
0
1
xyz
xa yb zc
Solution
1
\[\text{ a, b and c are in A . P }. \]
\[ \therefore 2b = a + c . . . . . . . . \left( i \right)\]
\[\text{ And, x, y and z are in G . P } . \]
\[ \therefore y^2 = xz\]
\[\text{ Now }, x^{b - c} y^{c - a} z^{a - b} \]
\[ = x^{b + a - 2b} y^{2b - a - a} z^{a - b} \left[ \text{ From } \left( i \right) \right]\]
\[ = x^{a - b} y^{2\left( b - a \right)} z^{a - b} \]
\[ = \left( xz \right)^{a - b} \left( xz \right)^{b - a} \left[ \text{ From } \left( ii \right), y^2 = xz \right]\]
\[ = \left( xz \right)^0 \]
\[ = 1\]
\[\]
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