SOLID GEOMETRY. I. Straight Line and Plane. 1994. The co-ordinates of four points are a-b, a-c, a-d; b − c, b-d, b-a; c-d, c-a, c-b; and d-a, d-b, d-c, respectively: prove that the straight line, joining the middle points of any two opposite edges of the tetrahedron of which they are the angular points, passes through the origin. 1995. Of the three acute angles which any straight line makes with three rectangular axes, any two are together greater than the third. 1996. The straight line joining the points (a, b, c), (a', b', c') will pass through the origin if aa + bb' + cc' = pp'; p, p' being the distances of the points from the origin, and the axes rectangular. Obtain the corresponding equation when the axes are inclined respectively at angles whose cosines are l, m, n. [aa' + bb' + cc' + (bc' + b’c) l + (ca' + c'a) m + (ab' + a’b) n = pp' √ 1 — l2 — m2 — n2 + 2lmn.] 1997. From any point P are drawn PM, PN perpendicular to the planes of zx, zy; O is the origin, and a, ß, y, the angles which OP makes with the co-ordinate planes and with the plane OMN: prove that cosec2 = cosec3 a + cosec2 ß + cosec2 y. 1998. The equations of a straight line are given in the forms. 1999. A straight line moves parallel to a fixed plane and intersects two fixed straight lines (not in one plane): prove that the locus of a point which divides the intercepted segment in a given ratio is a straight line. 2000. Determine what straight line is represented by the equations b + nx - lz a + mz - ny m-n (1) c + ly - mx c + ly - mx lb'-ma' ; [(1) The straight line at infinity in the plane x (m − n) + y (n − 1) + ≈ (1 − m) = 0 ; unless la + mb + nc = 0, in which exceptional case the line is indeterminate, and the locus of the equations is the plane x (m − n) + y (n − 1) + ż (l − m) = a+b+c; (2) the straight line at infinity in the plane x (mc' — nb') + y (na' — le') + z (lb′ — ma') = 0 ; unless la + mb + nc = 0, when the locus of the equations is the plane x (mc' — nb') + y (na' — lc') + z (lb′ — ma') = aa' + bb' + cc'.] 2002. The cosine of the angle between the two straight lines determined by the equations le + my + nz = 0, ax2 + by3 + cz2 = 0, l2 (b + c) + m2 (c + a) + n2 (a + b) is 2003. A straight line moves parallel to the plane y=≈ and intersects the curves (1) y=0, 2=cx; (2) ≈ = 0, y2=bx: z prove that the locus of its trace on the plane of yz is two straight lines. [The locus of the moving straight line is a = (y – ≈) ( 2004. The direction cosines of a number of fixed straight lines, referred to any system of rectangular axes, are (l,, m, n ̧), (1, m, n), &c. : prove that, if Σ (l3) = Σ (m2) = Σ (n2), and Σ(mn) = Σ (nl) = Σ (lm) = 0, when referred to one system of axes, the same equations will be true for any other system of rectangular axes. Also prove that, if these conditions be satisfied and a fixed plane be drawn perpendicular to each straight line, the locus of a point which moves so that the sum of the squares of its distances from the planes is constant will be a sphere having a fixed centre O which is the centre of inertia of equal particles at the feet of the perpendiculars drawn from O, and that the centre of inertia of equal particles at the feet of the perpendiculars drawn from any other point P lies on OP and divides OP in the ratio 2 : 1. 2005. A straight line always intersects at right angles the straight line x + y = z = 0, and also intersects the curve y=0, x = az: prove that the equation of its locus is represent in general three straight lines, two and two at right angles to gh hf fg b f =C they will represent a plane h meet the axis of x in 0, 0'; and points P, P' are taken on the two respectively such that (2) OP. O'P = c2; (3) OP + O'P' = 2c: (1) OP=k. O'P'; prove that the equation of the locus of PP' is (1) (x + a) (y sin a + z cos a) = k (x − a) (y sin a − ≈ cos a); the points being taken on the same side of the plane xy. [Denoting OP, O'P' by 2A, 2p, the = y-(λ-μ) cos a = equations of PP' may be written z-(+) sin a a (λ + μ) cos a (A) sin a so that, when any relation is given between λ, μ, immediately.] the locus may be found 2008. A triangle is projected orthogonally on each of three planes mutually at right angles: prove that the algebraical sum of the tetrahedrons which have these projections for bases and a common vertex in the plane of the triangle is equal to the tetrahedron which has the triangle for base and the common point of the planes for vertex. [This follows at once from the equation x cos a + y cos B + z cos y = p on multiplying both members by the area of the triangle.] prove that the two other straight lines in which it meets the surface - - (b − c) yz (mz – ny) + (c − a) zx (nx − lz) + (a − b) xy (ly – mx) = 0 are at right angles to each other. 2010. n : The direction cosines of three straight lines, which are two and two at right angles to each other, are (1 ̧, m, n,), (1,, m, n ̧), (m, n), and um ̧n, + bn,l, + cl ̧m1 = am ̧n ̧ + bnl ̧ + cl ̧m = 0: 2 2 = b 111 mmm 3 2011. The equations of the two straight lines bisecting the angles between the two given by the equations may be written l.x + my + nz = 0, ax2 + by2 + c~2 = 0, lx + my+nz = 0, 1 (b − c) yz + m (c − a) zx + n (a − b) xy = 0. 2012. The straight lines bisecting the angles between the two given by the equations lx + my + nz = 0, ax2 + by2 + cz2 + 2fy + 2gxx + 2hxy = 0, lie on the cone x (nh – mg) + + + yz {mh − ng + 1 (b − c)} + ... + = 0. 2013. The lengths of two of the straight lines joining the middle points of opposite edges of a tetrahedron are x, y, w is the angle between them, and a, a' the lengths of those edges of the tetrahedron which are not met by either x or y prove that 4xy cos w == a2 ~ a'2. 2014. The lengths of the three pairs of opposite edges of a tetrahedron are a, a'; b, b'; c, c': prove that, if be the acute angle between the directions of a and a', 2aa' cos 0 -- (b2 + b'3) ~ (c2 + c'2). 2015. The locus of a straight line which moves go as always to intersect the three fixed straight lines, is y=m (b− a), z = n (c − a); z = n (c − b), x = 1 (a−b); - x = 1 (a −c), y = m (b − c) ; lyz (b −c) + mzx (c − a) + nxy (a − b) – mnx (b − c)3 – ...... =2lmn (bc) (ca) (ab): and every such straight line also intersects the fixed line a (x-al) b(y-bm) c (z-en) m = n 2016. The straight line joining the centres of the two spheres, which touch the faces of the tetrahedron ABCD opposite to A, B respectively and the other faces produced, will intersect the edges CD, AB (produced) in points P, Q respectively such that CP : PD = ▲ACB : ▲ADB, and AQ: BQ=ACAD : ACBD. 2017. On three straight lines meeting in a point O are taken points A, a; B, b; C, c respectively prove that the intersections of the planes ABC, abc; aBC, Abc; AbC, aBc; and ABc, abC; all lie on one plane which divides each of the three segments harmonically to 0. 2018. Through any one point are drawn three straight lines each intersecting two opposite edges of a tetrahedron ABCD; and a, ƒ ; b, g; c, h are the points where these straight lines meet the edges BC, AD; CA, BD; AB, CD: prove that 2019. Any point O is joined to the angular points of a tetrahedron ABCD, and the joining lines meet the opposite faces in a, b, c, d : prove that Oa Ob Ос Od + + regard being had to the signs of the segments. Hence prove that the reciprocals of the radii of the eight spheres which can be drawn to touch the faces of a tetrahedron are the eight positive values of the expression P1 Pa Pa P, being the perpendiculars from the corners on the opposite faces. |