Prove the theorem of parallel axes.

Define the theorem of parallel axes and apply it to find the moment of inertia of a uniform rod about an axis passing through one of its ends and perpendicular to its length.

Prove the theorem of perpendicular axes. (Hint: Square of the distance of a pint (x,y) in the x - y plane from an axis perpendicular to the plane through the origin is x^2 + y^2 )

If a diameter of a cirlce bisects each of the chords of the circle, prove that the chordds are parallel.

If a transversal intersects two lines such that the bisectors of a pair of corresponding angles are parallel, then prove that the two lines are parallel.

Prove that the slope of a straight line is invariant under the translation of axes.

State Theorem of perpendicular axis.

E and F are the mid points of non-parallel sides AD and BC respectively of a trapezium prove that EF||AB.

(a) Prove that any line parallel to either of the asymptotes of a hyperbola shall meet it in one point at infinity. (b) Prove that the asymptotes of a hyperbola are the diagonals of the rectangle formed by the lines drawn parallel to the axes at the vertices of the hyperbola [i.e., at (pma, 0) and (0, pmb) ].