**Misc 7**

Last updated at Dec. 8, 2016 by Teachoo

Last updated at Dec. 8, 2016 by Teachoo

Transcript

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 = 𝑥2−𝑥12+𝑦2−𝑦12 Now, finding PS & PS’ PS’ = 𝑥−(−4)2+𝑦−02 PS’ = 𝑥+42+𝑦2 and PS = 𝑥−42+𝑦−02 PS = 𝑥−42+𝑦2 Now, PS + PS’ = 10 Putting values 𝑥−42+𝑦2 + 𝑥+42+𝑦2 = 10 𝑥−42+𝑦2 = 10 – 𝑥+42+𝑦2 Squaring both sides 𝑥−42+𝑦22 = 10 – 𝑥+42+𝑦22 𝑥−42+𝑦2 = 102 + 𝑥+42+𝑦22 – 20 𝑥+42+𝑦2 𝑥−42+𝑦2 = 100 + 𝑥+42+𝑦2 – 20 𝑥+42+𝑦2 x2 + 42 – 8x + 𝑦2 = 100 + x2 + 42 + 8x + 𝑦2 – 20 𝑥+42+𝑦2 x2 + 42 – 8x + 𝑦2 – 100 – x2 – 42 – 8x − 𝑦2 = – 20 𝑥+42+𝑦2 – 16x – 100 = – 20 𝑥+42+𝑦2 20 𝑥+42+𝑦2 = 16x + 100 20 𝑥+42+𝑦2 = 4(4x + 25) 5 𝑥+42+𝑦2 = (4x + 25) Squaring both sides 5 𝑥+42+𝑦22= (4x + 25)2 25𝑥+42+𝑦2 = (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 9𝑥2225 + 25𝑦2225 = 1 𝒙𝟐𝟐𝟓 + 𝒚𝟐𝟗 = 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 𝑥2𝑎2 + 𝑦2𝑏2 = 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 = 102 = 5m Also it is given that Distance between flag posts SS’ = 8m Distance between two foci = 8 m 2c = 8 c = 82 = 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 𝑥2𝑎2 + 𝑦2𝑏2 = 1 𝑥252 + 𝑦232 = 1 𝒙𝟐𝟐𝟓 + 𝒚𝟐𝟗 = 1 Thus, the required equation is 𝑥225 + 𝑦29 = 1

About the Author

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.