2008

1) Find the equation of the right circular cone generated by straight lines drawn from the origin to cut the circle that passes through the points (1,2,2), (2,1,−2) and (2,−2,1).

[10M]

2) Find the distance of the point (-2,3,-4) from the line x+23=2y+34=3z+45 measured parallel to the plane 4x+12y−3z+1=0.

[10M]

3) Obtain the equation of the sphere that touches the plane 3x+2y−z+2=0 at the point (1,−2,1) and cuts orthogonally the sphere x2+y2+z2−4x+6y+4=0.

[10M]

4) Derive the equations to the planes that touch the surface 4x2−5y2+7z2+13=0 and are parallel to the plane 4x +20y−21z=0. Also determine the coordinates of the points of contact.

[10M]

5) Consider the section of the enveloping cone of the ellipsoid x2a2+y2b2+z2c2=1 by the plane z=0. If the cone has its vertex at the point Pand if the section happens to be a rectangular hyperbola, show that the equation to the locus of the point P is x2+y2a2+b2+z2c2=1.

[10M]

2007

1) If the three concurrent lines whose direction cosines are (l1,mk,nj),(l2,m2,n2),(l3,m3,n3) are coplanar, prove that
[11m1n1l2m2n2l3m3n3]=0

[10M]

2) Find the equations of the three planes through the line

parallel to the axes.

[10M]

3) Prove that the shortest distance between the line

and any tangent to the ellipse x2sin2θ+y2=a2, z=0 is constant in length.

[10M]

4) The plane xa+yb+zc=1 cut the axes in A, B, C. Find the equation of the cone whose vertex is origin and the guiding curve is the circle ABC.

[10M]

5) Find the equation of the cylinder generated x1=y2=z3, the guiding curve being the conic z=2, 3x2+4xy+5y2=1.

[10M]

2006

1) Prove that the locus of a line which meets the two lines y=±mx,z=±c and the circle x2+y2=a2, z=0 is

[10M]

2) Two straight lines x−α1l1=y−β1m1=z−γ1n1;x−α2l2=y−β2m2=z−γ1n2 are cut by a third hine whose direction cosines are λ, u and v. Show that the length d intercepted on the third line is given by

d|l1m1n1l2m2n2λμν|= |α1−α2β1−β2γ1−γ2l1m1n1l2m2n2|

Deduce the length of the shortest distance between the first two lines.

[16M]

3) Find the condition that the plane lx+my+n=0 be a tangent plane to the cone ax2+by2+ cz2+2fyz+2gzx+2hxy=0.

[12M]

4) Prove that the locus of the pole of the plane lx+my+nz=p with respect to system of conicoids x2a2+k+y2b2+k+z2c2+k=1, where k is a parameter, is a straight line perpendicular to the given plane.

[12M]

2005

1) Find the equations of the generators of the hyperboloid x2a2+y2b2−z2c2=1 through any point of the principal elliptic section x2a2+y3b2−z2c2l; z=0.

[10M]

2) A variable plane is at a constant distance p from the origin and meets the axes in A,B and C Show that the locus of the centroid of the tetrahedron OABC is 1x2+1y2+1z2=16p2.

[10M]

3) Find the locus of the point of intersection of perpendicular generators of a hyperboloid of one sheet.

[10M]

4) Planes are drawn through a fixed point (α,β,γ) so that their sections of the paraboloid a2+by2=2z are rectangular hyperbolas. Prove that they touch the cone (x−α)2b+(y−β)2a+(z−γ)2a+b=0.

[10M]

5) Show that the enveloping cylinder of the conicoid
cx2+by2+cz7=1 with generators perpendicular to z-axis meets the plane z=0 in parabolas.

[10M]