PROBLEM III. Fig. 81. To draw a tangent plane to a cone through a given external point. Let v be the projections of the vertex, ABC the horizontal trace of the cone, and pp' the given point; it is required to draw through P a tangent plane to the cone. Construction. Find the traces L and N of the line VP; draw LM a tangent to ABC, and join MN. LMN is the plane required. Proof. Since the plane contains the line LN it passes through the points Pand V which are on that line; and as it contains LM, the tangent to ABC at D, and therefore the line VD, it is a tangent plane to the cone. Remarks. Should the vertical trace of PV be too remote, a point in the vertical trace of the plane may be found as in the last problem by drawing a line through V parallel to LM. As a second tangent to ABC may be drawn from L, this problem admits of two solutions. PROBLEM IV. Fig. 82. To draw a tangent plane to a cone which shall be parallel to a given straight line. It is required to draw a tangent plane to the cone VABC which shall be parallel to PQ. Construction. Find the traces of a line through V and parallel to PQ. These are L and N. Draw LM touching ABC and join MN. LMN is the plane required. Proof. Since LMN contains VN, a line parallel to PQ, LMN and PQ are parallel to one another (Theor. x. Ch. 1.), and as LMN contains the line VD and the tangent to ABC at the point D it is a tangent plane to the cone. Remark. It may be observed that when a tangent plane touches a surface along a line it can only be made to fulfil one other independent condition, such as passing through a given point, or being parallel to a given line. Thus a plane cannot generally be drawn to touch a cone and to contain a given line, for that would be equivalent to drawing a plane to contain two given lines, which is impossible except the lines are either intersecting or parallel. |