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If x,y,z,t are real numbers such that $x^{2} + y^{2} = 9, z^{2} + t^{2} = 4$ and $x t - y z = 6$ then the greatest value of xz is ________

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answer is A.

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Detailed Solution

let $\begin{array}{l} x = 3 \cos θ z = 2 \cos ϕ \\ y = 3 \sin θ t = 2 \sin ϕ \end{array}$
$\Rightarrow \sin (ϕ - θ) = 1 \Rightarrow ϕ - θ = 90^{\circ}$
$x z = - 6 \sin θ \cos θ = - 3 \sin 2 θ \leq 3$

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