Analytic Geometry — DU B.A. (Prog) Mathematics as Major Previous Year Question Papers

3. Attempt two parts from each question. (a) Sketch the graph of the parabola: y = 4x2 + 8x + 5, showing its vertex, focus and directrix. (b) Identity and sketch the curve x? + 4xy - 2y2 - 6 = 0. (c) Find the equation of the ellipse that satisfies the given conditions: foci (2, 1) and (2, - 3); major axis of length 6. (b) Sketch the following ellipse and label its foci, vertices, and ends of minor axis (c) State reflection property of the parabola. Find an equation of the parabola that satisfies the given conditions: focus (1, 1) and directrices y = - 2. 3. (a) Find the angle that the vector v = V3i + j makes with the positive x -axis. Also, find the vector whose length is 2 and same direction as v. (b) A vector w is a linear combination of the vectors V1, 22 and v3 if w can be expressed asw = GU, + C202 + C3V3, where Cy, Cz and C3 are scalars. Show that the vector 2i+ j- k cannot be expressed as a linear combination of 0, = i - j,02 = 3i + k and v3 = 4i-j+ k. (c) Express the vector vas the sum of a vector parallel to b and a vector orthogonal to b, where v = 4i- j + 7k,b = 2i + 3j- 6k. (a) Find the parametric equation of the line L passing through the points P, = (2,4, - 1) and P, - (5, 0, 7). Where does the line intersect the xy - plane? (b) Determine whether the line L: x = t, y = 2t, z = 3t and the plane x - y + 22 - 5 = O are parallel, perpendicular or neither. (c) Find the magnitude and the quations of the line of shortest distance between the lines: 5. (a) Find the equation of the sphere on the diameter with endpoints (1, - 2, 4) and (3.4, - 12) . Also, determine the center and the radius of the sphere. (b) Find the enveloping cone of the sphere x? +y? +22-2x + 4z - 1 = 0 and vertex (c) Find the equation of a right circular cylinder which envelopes a sphere with centre (a, b, c) and radius r and has the generators parallel to direction cosines (l, m, n). 6. (a) Find the equation of the sphere that touches the sphere x'+y2+2-x+3y+22-3=0 at the point (1,1, - 1) and passes through the origin. (b) Find the equation of the right circular cone with vertex at (1, -2, - 1) and axis is the line *→=*2=241 - and semi vertical angle is 60°. (c) Find the equation of the right circular cylinder of radius 2 whose axes passes through the point (1, 2, 3) and has the direction cosines proportional to (2,3, - 6).