Calculus With Analytic Geometry by SM. Yusaf & Prof.Muhammad Amin CHAPTER # 08 ANALYTIC GEOMETRY OF THREE DIMENSONS. Exercise #8.1

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1 CHAPTER # 08 ANALYTIC GEOMETRY OF THREE DIMENSONS Exercise #8.1 Show that the three given points are the vertices of a right triangle, or the vertices of an isosceles triangle, or both. Q#: A 1, 5, 0, B 6, 6,, C 0, 9, 5 Consider a ABC with vertices A 1,5,0, B 6,6,, C 0,9,5 For required proof, we have to find the following magnitudes AB = = = AB = BC = = = BC = 6 AC = = = AC = Since AB = AC Hence given triangle is isosceles. Q#5: A, 9,, B 0, 11,, C 1, 0, 1 Consider a ABC with vertices A,9,, B 0,11,, C 1,0,1 For required proof, we have to find the following magnitudes AB = = + + = AB = BC = = = BC = 1 AC = = = AC = 99 Since AB + AC = + 99 = 1 = 1 = BC AB + AC = BC Hence ABC is a right triangle with right angle at vertex A. Q#6: A 1, 0,, B,,, C 0, 7, 6 Do yourself as above Q#7: A,,, B 8, 1,, C, 1, 0 Do yourself as above Written and composed by M.Bilal (6500@gmail.com) Mardawal Naushehra,KHUSHAB Page 1

2 Q#8: Show that the points 1, 6, 1, 1,,, (,, 1) and (0,, 0) are the vertices of regular tetrahedron. Let given points are A = 1,6,1, B = 1,,, C =,,1 & D = 0,,0 to show that given points are the vertices of regular tetrahedron, for this we have to show that AB = AC = AD = BC = CD = BD AB = = = = 18 AB = AC = = = = 18 AC = AD = = = = 18 AD = BC = = = = 18 BC = CD = = = = 18 CD = BD = = = = 18 BD = Since AB = AC = AD = BC = CD = BD Hence proved the given points are the vertices of a regular tetrahedron. Q#9: Show that the points, 1,, 1, 1,, (, 1, 0) and (, 1, 1) are the vertices of rectangle. Suppose given points are A =, 1,, B = 1, 1,, C =,1,0, D =,1,1 For a rectangle, we have to show that AB = CD & BC = AD & A = 90 o AB = = AB = 5 CD = = CD = 5 BC = = = 9 BC = & AD = = = 9 AD = Hence AB = CD & BC = AD we have to prove A = 90 o Consider AB + AD = = (1) Since BD = BD = = 1 BD = 1 Putting in equation (1) AB + AD = BD So A = 90 o Hence given points are the vertices of rectangle. Written and composed by M.Bilal (6500@gmail.com) Mardawal Naushehra,KHUSHAB Page

3 Q#10: Under what conditions on x, y and z is the point P x, y, z equidistant from the points, 1, and 1, 5, 0? Suppose the given points are A, 1, and B 1,5,0 & Let P x, y, z be any point which is equidistance from A and B. According to given condition PA = PB x + y z = x y 5 + z 0 Square on both sides x + y z = x y 5 + z 0 x 6x y + y z 8z + 16 = x + x y 10y z 6x + y 8z + 6 = x 10y + 6 8x + 1y 8z = 0 x y + z = 0 x y + z = 0 is the required condition Q#11: Find the coordinates of the point dividing the join of A(, 1, ) and B(5, 1, 6) in the ratio : 5. Given points are A,1, & B(5, 1,6). Let P x, y, z be the required point dividing the line AB in ratio : 5 As we know that the P x, y, z divide the join of A x 1, y 1, z 1 & B x, y, z in the ratio m 1 : m is m x 1 + m 1 x m 1 + m, m y 1 + m 1 y m 1 + m, m z 1 + m 1 z m 1 + m Hence coordinates of point P are P , , P , , 8 8 P = 0,1, 19 is required point. Q#1: Find the ratio in which the yz-plane divides the segment joining the points A,, 7 and B, 5, 8. Given points are A,,7 & B, 5,8 Let the yz -plane divides the join of the given points in the ratio m 1 : m the x-coordinate of the point P dividing the join of given points in the ratio m 1 : m is x = m 1+ m m 1 +m = m 1 m m 1 +m Since this point lies on yz -plane so x = 0 m 1 m m 1 + m = 0 m 1 m = 0 m 1 = m m 1 m = m 1 : m = : is required ratio Written and composed by M.Bilal (6500@gmail.com) Mardawal Naushehra,KHUSHAB Page

4 Q#1: Show that the centroid of the triangle whose vertices are x i, y i, z i, i = 1,, ; is x 1 +x +x, y 1 +y +y, z 1+z +z. Let the given vertices of the triangle are A x 1, y 1, z 1, B x, y, z & C x, y, z. Suppose D, E, F are the mid points of the sides BC, AC & AB respectively. coordinates of D are D = x +x, y +y, z +z Suppose G is the centroid of ABC, Then coordinates of point G dividing AD in the ratio : 1 are G 1. x 1 + x + x 1 +, 1. y 1 + y + y 1 +, 1. z 1 + z + z 1 + G x 1 + x + x, y 1 + y + y, z 1 + z + z the coordinates of the points E and F are E x 1 + x, y 1 + y, z 1 + z & F x 1 + x, y 1 + y, z 1 + z Then the coordinates of the centroid G dividing BE in the ratio : 1 are G 1. x + x 1 + x 1 +, 1. y + y 1 + y 1 +, 1. z + z 1 + z 1 + G x 1 + x + x, y 1 + y + y, z 1 + z + z Similarly the coordinates of cancroids G dividing CF in the ratio : 1 are G x 1+x +x, y 1+y +y Hence coordinates of centroid G are G x 1+x +x, y 1+y +y, z 1+z +z, z 1+z +z Q#1: Find the centroid of the tetrahedron whose vertices are x i, y i, z i, i = 1,,,. Let the vertices of the tetrahedron are A = x 1, y 1, z 1 B = (x, y, z ) C = (x, y, z ) D = x, y, z Let E, F, G, H are the centroids of the triangle BCD, ACD, ABD & ABC respectively, Written and composed by M.Bilal (6500@gmail.com) Mardawal Naushehra,KHUSHAB Page

5 Then their coordinates are E = x + x + x, y + y + y, z + z + z F = x 1 + x + x G = x 1 + x + x H = x 1 + x + x, y 1 + y + y, y 1 + y + y, y 1 + y + y, z 1 + z + z, z 1 + z + z, z 1 + z + z coordinates of centroid dividing the line AE in ratio : 1 are 1. x 1 + x + x + x 1 +, 1. y 1 + y + y + y 1 +, 1. z 1 + z + z + z 1 + x 1 + x + x + x, y 1 + y + y + y, z 1 + z + z + z coordinates of centroid dividing the line BF in the ratio : 1 are 1. x + x 1 + x + x 1 +, 1. y + y 1 + y + y 1 +, 1. z + z 1 + z + z 1 + x 1 + x + x + x, y 1 + y + y + y, z 1 + z + z + z Similarly we can prove that co-ordinates of centroid in case of CG and DG are So co-ordinates of centroid are x 1 + x + x + x x 1 + x + x + x, y 1 + y + y + y, y 1 + y + y + y, z 1 + z + z + z, z 1 + z + z + z Checked by: Sir Hameed ullah ( gmail.com) Specially thanks to my Respected Teachers Prof. Muhammad Ali Malik (M.phill physics and Publisher of Muhammad Umar Asghar sb (MSc Mathematics) Hameed Ullah sb ( MSc Mathematics) Written and composed by M.Bilal (6500@gmail.com) Mardawal Naushehra,KHUSHAB Page 5

Downloaded from Class XI Chapter 12 Introduction to Three Dimensional Geometry Maths

Downloaded from Class XI Chapter 12 Introduction to Three Dimensional Geometry Maths A point is on the axis. What are its coordinates and coordinates? If a point is on the axis, then its coordinates and coordinates are zero. A point is in the XZplane. What can you say about its coordinate?