Prove that the lines
 and
 are perpendicular if

Find a, b, c, d so that the line
  pass through   and  and hence show that the given points and  are collinear.

Find the condition that the line
 intersect the conic z = 0, ax² + by² = 1. Hence show that the coordinates of any point on a line which intersects the conic and passes through the point  satisfy the equation
 =

Prove that the line  is parallel to the plane

Prove that the planes
 
 
 
pass through one line

Find the equation of the plane through the point  and the lines 
 
 
and find the equation to the lines through the given point which intersects the two given lines.

The plane  is rotated about the line of intersection with the plane  through an angle α. Find the equation of the plane in its new position.

The plane  cuts the axes in A, B, C. Find the equation of BC and the equation to the plane through O and perpendicular to BC. Prove that the three planes through O and perpendicular to BC, CA and AB respectively pass through a line.

Find the perpendicular distance of  from the line  

Find the image of the point  in the plane  

From a point  a line PM is drawn perpendicular to the line  and is produced to  such that . Show that
 
(l, m, n) are direction cosines

Find the equation to the two planes inclined at an angle α to the XY–plane and containing the line

Let P be a point on the plane . A point Q is taken on OP such that OP . OQ =p². Find the locus of Q.

Let L be the line
.
Find the direction cosines of the projection of L on the plane  and the equation of the plane through L and parallel to the line

Prove that the lines
 and
 
are coplanar if
 

Find the equation to the line which can be drawn from  to cut the lines
 
 

If no two of the three planes
 
are parallel then find the conditions under which these planes
i) intersect at a point
ii) they form a triangular prism
iii)intersect along a straight line

Prove that the planes
 
have a common line if lλ + mμ + nγ = 0. Find the distance of the line from the origin.

A variable plane makes with the coordinate plane a tetrahedron of constant volume 64k³. Find
i) the locus of the centroid of the tetrahedron
ii) the locus of the foot of the perpendicular from the origin to the plane.

If a > 0, b > 0 the maximum area of the triangle formed by the point O (0, 0), A (acosθ, bsinθ) and B (acosθ,bsinθ) is (in square units)

a)  when θ =

b)  when θ =

c)  when θ = 

d) a²b²

Let O be a fixed point and P any point on a given straight line; OP is joined and on it is taken a point Q such that OP . OQ = k². Prove that the locus of Q i.e. the inverse of the given straight line with respect to O is a circle which passes through O.

Let O be any point in the plane of a circle, and OP₁P₂ any chord of the circle which passes through O and meets the circle in P₁ and P₂. On this chord is taken a point Q such that OQ is equal to the arithmetic mean between OP₁ and OP₂. Find the locus of Q.