91Ó°ÊÓ

Given equation: \(x^2 + y^2 + 2z^2 + 2xz - 2yz = 1\) The coefficients of the equation are: - \(A = 1\) - \(B = 1\) - \(C = 2\) - \(F = -2\) - \(G = 2\) - \(H = 0\) Since \(A = B\), and \(H = 0\), the equation indeed represents a cylinder. Step 2: Determine the axis of the cylinder

The given equation can be written as: \(x^2 + y^2 - 2yz + 2xz + 2z^2 = 1\) Comparing to the general equation of the cylinder in the form \(Ax^2 + By^2 + Fyz + Gxz + Hxy + K = 0\), we can see that the cross terms are -2yz and 2xz, which represent the slopes for the lines parallel to the axis of the cylinder. Step 3: Determine the equation of the directrix

The coefficients of the cross terms in the equation give the direction ratios of the axis of the cylinder. Thus, direction ratios are: \(\alpha = 2, \beta = -2\), and the slope of the axis is \(\frac{\beta}{\alpha} = -1\). The directrix is a line parallel to the axis of the cylinder. We can write the equation of the directrix in the form: \(z = mx + ny + p\) As the axis has a slope of -1, we can write the equation of the directrix as: \(z = -x + p\) The equation of the directrix is not uniquely given, as the parameter \(p\) depends on the distance between the directrix and the point on the surface of the cylinder. We are only asked to find the equation of its directrix in general form: \(\boxed{z = -x + p}\)