I. Let us consider a fixed vertical line p and m be another line intersecting line p at a fixed point V and inclined to it at an angle α.

Question

11.Maths

I. Let us consider a fixed vertical line p and m be another line intersecting line p at a fixed point V and inclined to it at an angle α. Suppose we rotate the line m around the line p in such a way that the angle α remains constant then

1. What type of surface is generated and what do we call it?
2. What do we get if we take the intersection of this surface with a plane? What do we call them?
3. Do we get different types of curves depending upon the position of the intersecting plane?

II. Let β be the angle made by the intersecting plane with the vertical line p and the plane cuts the nappe either below or above the vertex. Which conic section do we obtain

1. When β = 90^o,
2.  When α < β < 90^o,
3.  When β = α,
4.  When 0 ≤ β < α;

III. In which of the above situations the plane cuts entirely across one nappe of the cone then

1. When the plane cuts through both the nappes? 
2. When do we get a circle?
3. Is the circle a conic section?

IV. For each of the following cases recognize the conic section and find the coordinates of the foci and vertices, the eccentricity, the axes of the conic section, the equations of the directrices and the length of the latus rectum.

1. y² = 10x 
2. 4x² + 9y² = 36 
3. 49y² – 16x² = 784

V. In the case of Ellipse find the lengths of major axis and minor axis and if the conic section is Hyperbola then find the lengths of transverse axis and conjugate axis. Draw graphs for each of the curves given above and analyse them.