38. Find the equation to a plane parallel to two given. straight lines; hence determine the shortest distance between two given straight lines.

39. Find the equation to a plane which passes through two given points and is perpendicular to a given plane.

40. There are n planes of which no two are parallel to each other, no three are parallel to the same right line, and no four pass through the same point; prove that the number of lines of intersection of the planes is 2(n-1), and that the number of points of intersection of those lines is

41. Find the shortest distance between the point (a, ß, y) and the plane

Ax+By+Cz = D.

42. Find the equation to the plane which passes through the origin and makes equal angles with three given straight lines which pass through the origin.

43. Determine the co-ordinates of the point which divides in a given ratio the distance between two points. Hence shew that the equation

Ax+By+Cz=D

must represent a plane, according to Euclid's definition of a plane.

44. Three planes meet in a point, and through the line of intersection of each pair a plane is drawn perpendicular to the third; prove that in general the planes thus drawn pass through the same line.

45. The equations to a line are

a-cy + bz B―az + cx

y-bx+ay

=

express them in the ordinary symmetrical form.

46. Find the condition that must subsist in order that the equations

a+mz

ny=0, b+nx – lz = 0,

c + ly — mx =

= 0

may represent a straight line; and supposing this condition to be satisfied put the equations in the ordinary symmetrical form.

47. Supposing the equations in the preceding question to represent a straight line, find in a symmetrical form the equations to the line from the origin perpendicular to the given line; also determine the co-ordinates of the point of intersection.

48. The locus of the middle points of all straight lines parallel to a fixed plane and terminated by two fixed right lines which do not intersect is a straight line.

49. The equation to a plane is lx+my+nz=0; find the equations to a line lying in this plane and bisecting the angle formed by the intersections of the given plane with the co-ordinate planes of (z, x) and (z, y).

50. A straight line, whose equations are given, intersects the co-ordinate planes in three points; find the angles included between the lines which join these points with the origin; and if these angles (a, ẞ, y) be given, shew that the equation to the surface traced out by the line in all positions is

x√(tan a) + y √(tan ẞ) + z √√(tan y) = 0.

II. Surfaces of the second order.

51. If the normal n at any point of an ellipsoid terminated in the plane of (x, y) make angles a, B, y with the semiaxes a, b, c, and p be the perpendicular from the centre on the tangent plane, then

n.p=c, and p2= a2 cos2 a + b2 cos2 B+ c2 cos2 y.

52. Find a point on an ellipsoid such that the tangent plane cuts off equal intercepts from the axes. Also find a point such that the intercepts are proportional to the axes.

53. If a, b, c be the semiaxes of an ellipsoid taken in order, and e, e' the excentricities of the principal sections containing the mean axis, shew that the perpendiculars from the centre on the tangent planes at every point of the section of the surface made by the plane abe'zc'ex are equal.

54. From a given point O, a line OP is drawn meeting a given plane in Q, and the rectangle OP. OQ is invariable; find the locus of P.

55. Sections of an ellipsoid are made by planes which all contain the least axis; find the locus of the foci of the sections.

56. Find the locus of a point which is equidistant from every point of the circle determined by the equations

x2 + y2+z2 = a2,

lx +my+nz=

p.

57. Shew that the section of the surface z2= xy by the plane z = x+y+c is a circle.

58. Interpret the equation

x2 + y2+ z2 = (lx + my + nz)2.

59. A, B, C are three fixed points, and P a point in space such that PA+ PB2 = PC2; find the locus of P, and explain the result when ACB is a right or obtuse angle.

are in order of magnitude, may be written thus,

Te2 (x2 + y2+ z2 — b2) — x2 + m2x2 = 0,

where k and m are certain constants. Hence shew that two circular sections of an ellipsoid can be obtained by cutting it by planes passing through its mean axis.

61. A sphere (C) and a plane are given; shew that if any sphere (C) be described touching the plane at a given point and cutting C, the plane of section always contains a given right line. Shew also that if the point of contact be not given, and if the plane of section always contain a given point, the centre of the sphere C' will always be upon a given paraboloid.

62. If A, B, C be extremities of the axes of an ellipsoid, and AC, BC be the principal sections containing the least axis, find the equations to the two cones whose vertices are A, B, and bases BC, AC respectively; shew that the cones have a common tangent plane, and a common parabolic section, the plane of the parabola and the tangent plane intersecting the ellipsoid in ellipses, the area of one of which is double that of the other; and if I be the latus rectum of the parabola, 1, 1⁄2 of the sections AC, BC, prove that

63. Tangent planes are drawn to an ellipsoid from a given external point; find the equation to the cone which has its vertex at the origin and passes through all the points of contact of the tangent planes with the ellipsoid.

64. If tangent planes be drawn to an ellipsoid from any point in a plane parallel to that of (x, y), the curve which contains all the points of contact will lie in a plane which always cuts the axis of z in the same point.

65. Shew that the tangent plane to an ellipsoid is expressed by the equation

lx + my + nz = √(l2a2 + m2b2 + n2c2).

66. Form the equation to the plane which passes through a given point of an ellipsoid, through the normal at that point, and through the centre of the ellipsoid.

67. A line passing through a given point moves so that the projection of any portion of it on a given line bears a constant ratio to the length of that portion; find the equation to the surface which it traces out.

68. Find the length of the perpendicular from a given point on a given straight line in space.

Investigate the equation to a right cone, having the axis, vertex, and vertical angle given; and determine the condition under which the section made by a plane parallel to one of the co-ordinate planes will be an ellipse.

69. Determine the radii of the spheres which touch the co-ordinate planes and the plane x+y+z=h.

70. An ellipsoid is intersected in the same curve by a variable sphere, and a variable cylinder; the cylinder is always parallel to the least axis of the ellipsoid, and the centre of the sphere is always at the focus of a principal section containing this axis. Prove that the axis of the cylinder is invariable in position, and that the area of its transverse section varies as the surface of the sphere.

71. Three edges of a tetrahedron, in length a, b, and c, are mutually at right angles; prove that if these three edges be taken as axes, the equation to the cone which has the origin for vertex, and for its base the circle circumscribed about the opposite face, is

(C + i) y= + (2 + 2) 2x + († + 1 ) xy = 0,

(+)

and that the plane ax+by+cz=0 is parallel to the subcontrary sections of the cone.

Find the corresponding equations when either of the other angular points of the tetrahedron is taken as vertex.