If $a, b, c$ are in A.P., $a, x, b$ are in G.P. and $b, y, c$ are in G.P., the point $(x, y)$ lies on

x^{2} + y^{2} = 2 b^{@}

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x^{2} - y^{2} = 2 b^{2}

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x^{2} - y^{2} = 3 b^{2}

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x^{2} + y^{2} = 3 b^{2}

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Solution

The correct option is D $x^{2} + y^{2} = 3 b^{2}$
$a, b, c$ are in $A \cdot P$
$a, x, b$ are in G.P
$b, y, c$ are in G.P
From above,
$2 b = a + c - (i)$
$x^{2} = a b - (i i)$
$y^{2} = b c - (i i i)$
On adding (ii) and (iii)
$x^{2} + y^{2} = (a + c) b$
$\Rightarrow x^{2} + y^{2} = 2 b^{2}$ (from (i))

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