Ex 12.2, 5 - Find equation of set of points P, the sum - Distance between two points - Set of points

  1. Chapter 12 Class 11 Introduction to Three Dimensional Geometry
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Ex 12.2, 5 Find the equation of the set of points P, the sum of whose distances from A (4, 0, 0) and B (–4, 0, 0) is equal to 10. Given A (4, 0, 0) B ( – 4, 0, 0) Let the Co-ordinate of point P be (x, y, z) Given that , Sum Distance of PA & PB is 10 i.e. PA + PB = 10 First, lets calculate PA & PB Calculating PA P (x, y, z) A (4, 0, 0) PA = ﷐﷮﷐x2−x1﷯2+﷐y2−y1﷯2+﷐z2 −z1﷯2﷯ Here, x1 = x, y1 = y, z1 = z x2 = 4, y2 = 0, z2 = 0 PA = ﷐﷮﷐4−𝑥﷯2+﷐0−𝑦﷯2+﷐0−𝑧﷯2﷯ = ﷐﷮﷐4−𝑥﷯2+﷐−𝑦﷯2+﷐−𝑧﷯2﷯ = ﷐﷮﷐4−𝑥﷯2+𝑦2+𝑧2﷯ Calculating PB P (x, y, z) B ( – 4, 0, 0) PB = ﷐﷮﷐x2−x1﷯2+﷐y2−y1﷯2+﷐z2 −z1﷯2﷯ Here, x1 = x, y1 = y, z1 = z x2 = – 4, y2 = 0, z2 = 0 PB = ﷐﷮﷐−4−𝑥﷯2+﷐0−𝑦﷯2+﷐0−𝑧﷯2﷯ = ﷐﷮﷐−1﷯2﷐4+𝑥﷯2+﷐−𝑦﷯2+﷐−𝑧﷯2﷯ = ﷐﷮﷐4+𝑥﷯2+𝑦2+𝑧2﷯ Now, given that PA + PB = 10 Putting value of PA & PB ﷐﷮﷐4−𝑥﷯2+y2+z2﷯ + ﷐﷮﷐4+𝑥﷯2+y2+z2﷯ = 10 ﷐﷮﷐4−𝑥﷯2+y2+z2﷯ = 10 – ﷐﷮﷐4+𝑥﷯2+y2+z2﷯ Squaring both sides ﷐4−𝑥﷯2+y2+z2 = ﷐﷐10−﷐﷮﷐4+𝑥﷯2+y2+z2﷯﷯﷮2﷯ (4 – x)2 + y2 + z2 = (10)2 + ﷐﷐﷮﷐4−𝑥﷯2+y2+z2﷯﷯2 – 2 (10) (4 – x)2 + y2 + z2 = 100 + (4 + x)2 + y2 + z2 – 20﷐﷮﷐4+𝑥﷯2+y2+(z)2﷯ (4 – x)2 –(4 + x)2 + y2 – y2 + z2 – z2 = – 20﷐﷮﷐4+𝑥﷯2+y2+z2﷯ (4 – x)2 – (4 + x)2 + 0 + 0 – 100 = – 20﷐﷮﷐4+𝑥﷯2+y2+z2﷯ ((4 – x) – (4 + x))((4 – x) + (4 + x)) – 100 = – 20﷐﷮﷐4+𝑥﷯2+y2+z2﷯ (4 – x – 4 + x)(4 – x + 4 + x) – 100 = – 20﷐﷮﷐4+𝑥﷯2+y2+z2﷯ ( – 2) (8) – 100 = – 20﷐﷮﷐4+𝑥﷯2+y2+z2﷯ – 16x – 100 – 20 ﷐﷮﷐4+𝑥﷯2+y2+z2﷯ – 4 (4x + 25) = – 4(5) ﷐﷮﷐4+𝑥﷯2+y2+z2﷯ (x + 25) = – 5﷐﷮﷐4+𝑥﷯2+y2+z2﷯ Again Squaring both sides (x + 25)2 = ( – 5)2 ﷐﷐﷮﷐4+𝑥﷯2+y2+z2﷯﷯2 (x + 25)2 = 25 ((4 + x)2 + y2 + z2) (4x + 25)2 = 25 ((4 + x)2 + y2 + z2) (4x)2 + (25)2 + 2(25)(x) = 25 ((4)2 + x2 + 2(4)(x) + y2 + z2 ) 16x2 + 625 + 200 x = 25(16 + x2 + 8x + y2 + z2 ) 16x2 + 625 + 200 x = 400 + 25x2 + 200x + 25y2 + 25z2 16x2 + 200 x – 25x2 – 200 x –25y2 – 25z2 = 400 – 625 16x2 – 25x2 – 25y2 – 25z2 – 200x + 200x = 225 – 9x2 –25x2 – 25y2 + 0 = 225 0 = 9x2 + 25x2 + 25y2 + 225 9x2 + 25x2 + 25y2 + 225 = 0 Which is the required equation of set of points P(x,y,z)

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