Find the locus of the middle points of the normal chord of the ellipse

Prove that the product of a focal distance of a point P is equal to the square of the semi diameter parallel to the tangent at P.

If PCP' and DCD' be a pair of conjugate diameters then show that
 i)  is constant, and
ii) the area of the parallelogram formed by the tangents at the end of diameters is constant.

Let P : (x´, y´) be point on the ellipse and the tangent and normal meet the axis in T and G respectively and let PN be the ordinate of P. If S be the focus of the ellipse then and the tangent and normal at P bisect the external and internal angles between the focal distances of P.

Let S and S´ be the two foci of an ellipse and  be perpendiculars from the foci upon the tangents at any point P of the ellipse, then show that Y and Y´ lie on the auxiliary circle, and . Also show that  are parallel.

Show that if the normals at any point P on the ellipse  meet the major and the minor axis in G and g, and if CF is the perpendicular upon the normal from the centre C, then  and

If p be the distance of the origin from a line whose direction cosines are (l, m, n) and p₁, p₂, p₃ are the distances of the origin from the projections of the line on the three co-ordinate planes respectively, then show that
 

If a variable plane moves in such a way that the sum of the reciprocals of its intercepts on the axes is constant, then prove that the plane passes through a fixed point.

Find the constant λ, so that the planes  and  are right angles and find in that case plane through the points  and perpendicular to both the given planes.

a) -8,  

b) -4, 

c) -6, 

d) 8, 

A plane passes through the line of intersection of the planes  and. Its perpendicular distance from the origin is unity. Show that its equation is given by

Note – The given planes may have been given as x + 3y + 6 = 0 and 3x – y – 4z = 0.

Show that the point of intersection of the line  r x b = c and the plane r . n = p is given by  

Show that the points of intersection of the straight line
  and the surface 11x² – 5y² + z² = 0 are (1, 2, 3) and (2, -3, 1).

Show that the distance of the point  from the point of intersection of line  and the plane
  is 6.

Show that the distance of the point a from the plane r . n = d measured parallel to the straight line r x m = 0 is   

Show that the distance of the point  from the plane  measured parallel to the line  is 1.

Show that the equation of the line through a which is perpendicular to the lines r x m₁ = 0 and r x m₂ = 0 is given by

Show that the equation of the line of intersection of the planes r . n₁ = p₁ and r . n₂ = p₂ is given by 

Find the angle between the lines
 
 

Assuming the plane   to be horizontal, find the inclination of the plane  to the horizontal.

Assuming the plane  to be horizontal, find the equation of the line of the greatest slope in the above plane passing through

Assuming the plane  to be horizontal, find the equation of the horizontal line through  which lies in the plane

Assuming the plane  to be horizontal, find the equation of the vertical plane passing through  which contains the line of greatest slope

Assuming the plane  to be horizontal, find the inclination of the line lying in the plane  and making an angle 45° with the line of greatest slope in the plane, with the horizontal.