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Example 12 - Find equation of set of points P such that distances - Distance between two points - Set of points

  1. Chapter 12 Class 11 Introduction to Three Dimensional Geometry
  2. Serial order wise
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Example 12 Find the equation of the set of the points P such that its distances from the points A (3, 4, –5) and B (– 2, 1, 4) are equal. Given A (3, 4, 5) & B ( –2, 1, 4,) Let point P be (x, y, z,) Given PA = PB Calculating PA P (x, y, z) A (3, 4, – 5) PA = ﷐﷮﷐x2−x1﷯2+﷐y2−y1﷯2+﷐z2 −z1﷯2﷯ Here, x1 = x, y1 = y, z1 = z x2 = – 2, y2 = 1, z2 = 4 PA = ﷐﷮﷐3−𝑥﷯2+﷐4−𝑦﷯2+﷐−5−𝑍﷯2﷯ = ﷐﷮﷐3−𝑥﷯2+﷐4−𝑦﷯2+﷐5+𝑍﷯2﷯ = ﷐﷮﷐3﷯2+﷐𝑥﷯2−2﷐3﷯﷐𝑥﷯+﷐4﷯2+𝑦2−2×4﷐−𝑦﷯+﷐5﷯2+﷐𝑧﷯2+2(5)(𝑧) ﷯ = ﷐﷮9+𝑥2−6𝑥+16+𝑦2−8𝑦+25+𝑧2+10𝑧﷯ = ﷐﷮𝑥2+𝑦2+𝑧2−6𝑥−8𝑦+10𝑧 9+16+25﷯ = ﷐﷮𝑥2+𝑦2+𝑧2−6𝑥−8𝑦+10𝑧+40﷯ Calculating PB P (x, y, z) B ( –2, 1, 4) PB = ﷐﷮﷐x2−x1﷯2+﷐y2−y1﷯2+﷐z2 −z1﷯2﷯ Here, x1 = x, y1 = y, z1 = z x2 = – 2, y2 = 1, z2 = 4 PB = ﷐﷮﷐−2−𝑥﷯2+﷐1−𝑦﷯2+﷐4−𝑧﷯2﷯ = ﷐﷮﷐2+𝑥﷯2+﷐1−𝑦﷯2+﷐4−𝑧﷯2﷯ = ﷐﷮﷐2﷯2+﷐𝑥﷯2+2﷐2﷯﷐𝑥﷯+﷐1﷯2+𝑦2−2﷐1﷯﷐𝑦﷯+42+𝑧2−2(4)(𝑧) ﷯ = ﷐﷮4+𝑥2+4𝑥+1+𝑦2−2𝑦+16+𝑧2−8𝑧﷯ = ﷐﷮𝑥2+𝑦2+𝑧2−4𝑥−2𝑦+8𝑧+21﷯ = ﷐﷮𝑥2+𝑦2+𝑧2−6𝑥−8𝑦+10𝑧+40﷯ Now ,given that PA = PB ﷐﷮𝑥2+𝑦2+𝑧2−6𝑥−8𝑦+10𝑧+40﷯ = ﷐﷮𝑥2+𝑦2+𝑧2+4𝑥−2𝑦+8𝑧+21﷯ Squaring both sides ﷐﷐﷮𝑥2+𝑦2+𝑧2−6𝑥−8𝑦+10𝑧+40﷯﷯2 = ﷐﷐﷮𝑥2+𝑦2+𝑧2+4𝑥−2𝑦+8𝑧+21﷯﷯2 𝑥2+𝑦2+𝑧2−6𝑥−8𝑦+10𝑧+40 = 𝑥2+𝑦2+𝑧2+4𝑥−2𝑦+8𝑧+21 𝑥2+𝑦2+𝑧2−6𝑥−8𝑦+10𝑧+40 – 𝑥2−𝑦2−𝑧2+4𝑥+2𝑦+8𝑧−21=0 𝑥2−𝑥2+𝑦2+𝑦2+𝑧2−𝑧2−6𝑥−4𝑥+8𝑦+2y+10z+8z+40−21=0 0 + 0 + 0 – 10x – 6y + 18z + 29 = 0 – 10x – 6y + 10z + 29 = 0 0 = 10x + 6y – 10z – 29 = 0 10x + 6y – 10z – 29 = 0 which is the required equation

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