SINGULAR POINTS Since at (0, 0), from (^4), dx 2 dxdy = Xe, it is evident that (D) may be written in the form (*> dxdydx In the same manner, if
there is a triple point at the origin, the equation of the three tangents bein g fz* + gx*y + hxy* + iy* = 0, and so on in general. If we wish to investigate the appearance of a curve at a given point, it is of fundamental importance to solve the tangent problem for that point. The above results indicate that this can be done by simple inspection after we have transformed the origin to that point. Hence we have the following rule for finding the tangents at a given point. FIRST STEP. Transform the origin to the point in question. SECOND STEP. Arrange the terms of the resulting equation according to ascending powers of x and y. THIRD STEP. Set the group of terms of lowest degree equal to zero, This gives the equation of the tangents at the point (origin). ILLUSTRATIVE EXAMPLE 1. Find the equation of the tangent to the ellipse x 2 + 5y 2 + 2xy-12x-12y = at the origin. Solution. Placing the terms of lowest (first) degree equal to zero, we get x 12y = 0, or x + y = 0, which is then the equation of the tangent P Tat the origin. ILLUSTRATIVE EXAMPLE 2. Examine the curve 3 x 2 xy 2-y* + x 3 8 y 3 = for tangents at the origin. Solution. Placing the terms of lowest (second) degree equal to zero,
x 2 - xy - 2 y 2 = 0,
or (x-y)(3x + 2y) = 0, x y = being the equation of the tangent AB, and 3x + 2y = the equation of the tangent CD. The origin is, then, a double point of the curve.
