[1] Start from the hyperbolic paraboloid surface z = f(x, y) = y 2x 2 , and verify that the point P defined by (1, 2, 3) lies on it.

[2] Find the scalar equation of the tangent plane at P, and write as a function z = T(x, y).

[3] Equating f(x, y) = T(x, y) gives a quadratic expression for (x, y) values where the hyperbolic paraboloid surface intersects the tangent plane. Completing the squares in this expression leads to a difference of squares that nicely factors into the form P1(x, y)P2(x, y) = 0.

[4] Choose one of these factors to be zero, say P1(x, y) = 0. It is the equation of a plane. The line of intersection of this plane with the tangent plane z = T(x, y) gives one of these magical lines construct the parametric equations for this line. (Hints: see example #7 in Section 12.5, and point P is on the line.)

[5] (Optional) Verifying that every point on this line lies on the surface z = y 2 x 2 checks your math.