PYQs

1) Find the equations of the two generating lines through any point (acosθ,bsinθ,0) of the principal elliptic section x2a2+y2b2=1, z=0 of the hyperboloid by the plane z=0.

[2014, 15M]

2) A variable generator meets two generators of the system through the extremities B and B′ of the minor axis of the principal elliptic section of the hyperboloid x2b2−z2c2=1 in P and P′ prove that BP⋅P′B′=a2+c2.

[2013, 20M]

3) Show that generators through any one of the ends of an equi-conjugate diameter of the principal elliptic section of the hyperboloid x2a2+y2b2−z2c2=1 are inclined to each other at an angle of 60∘ if a2+b2=6c2. Find also the condition for the generators to be perpendicular to each other.

[2011, 20M]

4) Find the vertices of the skew quadrilateral formed by the four generators of the hyperboloid x24+y2−z2=49 passing through (10,5,1) and $(14,2,-2)$$.

[2010, 20M]

5) Show that the equation x2−5xy+y2+8x−20y+15=0 represents a hyperbola. Find the coordinates of its center and the length of its real semi-axes.

[2001, 12M]