A Note in Analytical Geometry

If the equation to a straight line be

lx + my + n = 0

we know that the equation of the diameter conjugate to the line with respect to the conic

f(x, y) = ax² + 2hxy + by² + 2gx + 2fy + c = 0

is X/l = Y/m, or mX - lY = 0,

where X = ax + hy + g, Y = hx + by + f.

In other words, the diameter conjugate to a given line is obtained by substituting (- Y, X) for (x, y) in the original equation and suppressing the constant.