Let P be an arbitrary point having sum of the squares of the distance from the planes x+y+z=0, lx-nz=0 and x-2y+z=0 equal to 9. If the locus of the point P is x² + y² + z² = 9, then the value of l - n is equal to ______.

<p>The problem provides an equation involving the coordinates (α, β, γ) of a point P:<br>((α + β + γ) / √3)^2 + ((α - nγ) / √(l^2 + n^2))^2 + ((α - 2β + γ) / √6)^2 = 9</p><p>The locus of P(α, β, γ) is given by replacing (α, β, γ) with (x, y, z):<br>((x + y + z) / √3)^2 + ((lx - nz) / √(l^2 + n^2))^2 + ((x - 2y + z) / √6)^2 = 9</p><p>This represents the equation of an ellipsoid. The text proceeds by comparing coefficients. By expanding the equation, the coefficients of x^2, y^2, z^2, and cross-product terms are collected. From the given conditions:<br>Coefficient of x^2: 1/3 + l^2 / (l^2 + n^2) + 1/6 = 1<br>Coefficient of y^2: 1/3 + 0 + 4/6 = 1<br>Coefficient of z^2: 1/3 + n^2 / (l^2 + n^2) + 1/6 = 1<br>Coefficient of xz: -2ln / (l^2 + n^2) + 2/6 = 0<br>(The document seems to simplify this differently).</p><p>The text provides a shortcut:<br>1/3 + (l^2 / (l^2 + n^2)) + 1/2 = 1 (Combining 1/3 and 1/6 to 1/2)<br>And also, (ln / (l^2 + n^2)) = 0. This implies ln = 0.</p><p>If ln = 0, then from l^2/(l^2+n^2) = 1/2, this means 2l^2 = l^2+n^2, so l^2 = n^2, which implies l = ±n.<br>The text states l^2 + n^2 - 2ln = 0, which simplifies to (l - n)^2 = 0, therefore l - n = 0 or l = n.</p>

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