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Misc 7 - A man running a racecourse notes that sum of distances - Ellipse - Path traced

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  1. Chapter 11 Class 11th Conic Sections
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Misc 7(Method 1) A man running a racecourse notes that the sum of the distances from the two flag posts from him is always 10 m and the distance between the flag posts is 8 m. find the equation of the posts traced by the man. Let man be standing on point P(x, y) There are two flag posts S & S’ Given PS + PS’ = 10 & SS’ = 8 m Let S & S’ be on x-axis such that Origin (O) be the mid-point of S’S So, OS = OS’ = 4 m So, coordinates of S = S(4, 0) coordinates of S’ = S’(–4, 0) We know that distance between two points (x1, y1) & (x2, y2) d = ﷐﷮﷐﷐﷐𝑥﷮2﷯−﷐𝑥﷮1﷯﷯﷮2﷯+﷐﷐﷐𝑦﷮2﷯−﷐𝑦﷮1﷯﷯﷮2﷯﷯ Now, finding PS & PS’ PS’ = ﷐﷮﷐﷐𝑥−(−4)﷯﷮2﷯+﷐﷐𝑦−0﷯﷮2﷯﷯ PS’ = ﷐﷮﷐﷐𝑥+4﷯﷮2﷯+﷐𝑦﷮2﷯﷯ and PS = ﷐﷮﷐﷐𝑥−4﷯﷮2﷯+﷐﷐𝑦−0﷯﷮2﷯﷯ PS = ﷐﷮﷐﷐𝑥−4﷯﷮2﷯+﷐𝑦﷮2﷯﷯ Now, PS + PS’ = 10 Putting values ﷐﷮﷐﷐𝑥−4﷯﷮2﷯+﷐𝑦﷮2﷯﷯ + ﷐﷮﷐﷐𝑥+4﷯﷮2﷯+﷐𝑦﷮2﷯﷯ = 10 ﷐﷮﷐﷐𝑥−4﷯﷮2﷯+﷐𝑦﷮2﷯﷯ = 10 – ﷐﷮﷐﷐𝑥+4﷯﷮2﷯+﷐𝑦﷮2﷯﷯ Squaring both sides ﷐﷐﷐﷮﷐﷐𝑥−4﷯﷮2﷯+﷐𝑦﷮2﷯﷯﷯﷮2﷯ = ﷐﷐10 – ﷐﷮﷐﷐𝑥+4﷯﷮2﷯+﷐𝑦﷮2﷯﷯﷯﷮2﷯ ﷐﷐𝑥−4﷯﷮2﷯+﷐𝑦﷮2﷯ = 102 + ﷐﷐﷐﷮﷐﷐𝑥+4﷯﷮2﷯+﷐𝑦﷮2﷯﷯﷯﷮2﷯ – 20 ﷐﷮﷐﷐𝑥+4﷯﷮2﷯+﷐𝑦﷮2﷯﷯ ﷐﷐𝑥−4﷯﷮2﷯+﷐𝑦﷮2﷯ = 100 + ﷐﷐𝑥+4﷯﷮2﷯+﷐𝑦﷮2﷯ – 20 ﷐﷮﷐﷐𝑥+4﷯﷮2﷯+﷐𝑦﷮2﷯﷯ x2 + 42 – 8x + ﷐𝑦﷮2﷯ = 100 + x2 + 42 + 8x + ﷐𝑦﷮2﷯ – 20 ﷐﷮﷐﷐𝑥+4﷯﷮2﷯+﷐𝑦﷮2﷯﷯ x2 + 42 – 8x + ﷐𝑦﷮2﷯ – 100 – x2 – 42 – 8x − ﷐𝑦﷮2﷯ = – 20 ﷐﷮﷐﷐𝑥+4﷯﷮2﷯+﷐𝑦﷮2﷯﷯ – 16x – 100 = – 20 ﷐﷮﷐﷐𝑥+4﷯﷮2﷯+﷐𝑦﷮2﷯﷯ 20 ﷐﷮﷐﷐𝑥+4﷯﷮2﷯+﷐𝑦﷮2﷯﷯ = 16x + 100 20 ﷐﷮﷐﷐𝑥+4﷯﷮2﷯+﷐𝑦﷮2﷯﷯ = 4(4x + 25) 5 ﷐﷮﷐﷐𝑥+4﷯﷮2﷯+﷐𝑦﷮2﷯﷯ = (4x + 25) Squaring both sides ﷐﷐5 ﷐﷮﷐﷐𝑥+4﷯﷮2﷯+﷐𝑦﷮2﷯﷯﷯﷮2﷯= (4x + 25)2 25﷐﷐﷐𝑥+4﷯﷮2﷯+﷐𝑦﷮2﷯﷯ = (4x + 25)2 25(x2 + 42 + 8x + y2) = (4x)2 + (25)2 + 2(4x)(25) 25x2 + 400 + 200x + 25y2 = 16x2 + 625 + 200x 9x2 + 25y2 = 225 ﷐9﷐𝑥﷮2﷯﷮225﷯ + ﷐25﷐𝑦﷮2﷯﷮225﷯ = 1 ﷐﷐𝒙﷮𝟐﷯﷮𝟐𝟓﷯ + ﷐﷐𝒚﷮𝟐﷯﷮𝟗﷯ = 1 Misc 7(Method 2) A man running a racecourse notes that the sum of the distances from the two flag posts form him is always 10 m and the distance between the flag posts is 8 m. find the equation of the posts traced by the man. Let S & S’ be two flag past The sum of distance of man P from S & S’ is equal to 10 Since the sum of distance of a point from any two fixed points S & S’ in the plane is constant, it forms on ellipse. So, S & S’ are the foci of the ellipse x-axis is the major axis, y-axis is the minor axis Let equation of ellipse be ﷐﷐𝑥﷮2﷯﷮﷐𝑎﷮2﷯﷯ + ﷐﷐𝑦﷮2﷯﷮﷐𝑏﷮2﷯﷯ = 1 Now, given that The sum of distance of man P from S & S’ = 10 i.e. Sum of distance of a point from foci is = 10 2a = 10 m a = ﷐10﷮2﷯ = 5m Also it is given that Distance between flag posts SS’ = 8m Distance between two foci = 8 m 2c = 8 c = ﷐8﷮2﷯ = 4 Now we need to find b We know that c2 = a2 − b2 42 = 52 − b2 16 = 25 − b2 b2 = 25 −16 b2 = 9 b = 3 Now equation of ellipse is ﷐﷐𝑥﷮2﷯﷮﷐𝑎﷮2﷯﷯ + ﷐﷐𝑦﷮2﷯﷮﷐𝑏﷮2﷯﷯ = 1 ﷐﷐𝑥﷮2﷯﷮﷐5﷮2﷯﷯ + ﷐﷐𝑦﷮2﷯﷮﷐3﷮2﷯﷯ = 1 ﷐﷐𝒙﷮𝟐﷯﷮𝟐𝟓﷯ + ﷐﷐𝒚﷮𝟐﷯﷮𝟗﷯ = 1 Thus, the required equation is ﷐﷐𝑥﷮2﷯﷮25﷯ + ﷐﷐𝑦﷮2﷯﷮9﷯ = 1

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can contact him here.
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