Answer Happy

Question 3. Finding tangent planes through certain "anchors" and certain directions: (a) Find all planes which (i) are tangent to the elliptic paraboloid z = x² + y², and (ii) pass through both points P= (0, 0, -1) and Q = (2,0,3). How many such planes are there? (b) Find all planes which (i) are tangent to the surface z = x + xy² - y³, (ii) are parallel to the vector = (3, 1, 1), and (iii) pass through the point P = (-1, -2, 3). How many such planes are there? (c) Find all planes which (i) are tangent to the surface z = x² + sin y, (ii) are parallel to the x-axis, and (iii) pass through the point P = (0,0,-5). How many such planes are there?