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Misc 4 - An arch is in form of semi ellipse. It is 8 m wide - Ellipse - Path traced

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  1. Chapter 11 Class 11 Conic Sections
  2. Serial order wise
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Misc 4 An arch is in the form of semi ellipse. It is 8 m wide and 2 m high at the centre. Find the height of the arch at a point 1.5 m from one end. The arch is in form of semi−ellipse It is 8m wide & 2m high ⇒ Let AB = width of semi-ellipse = 8 m & CO = Height of semi-ellipse = 2 m We know that Length of Major axis of ellipse = 2a AB = 2a 8 = 2a 2a = 8 a = ﷐𝟖﷮𝟐﷯ = 4 Now, equation of ellipse ﷐𝑥2﷮𝑎2﷯ + ﷐𝑦2﷮𝑏2﷯ = 1 Putting values of a & b ﷐𝑥2﷮﷐4﷮2﷯﷯ + ﷐𝑦2﷮﷐2﷮2﷯﷯ = 1 ﷐𝒙𝟐﷮𝟏𝟔﷯ + ﷐𝒚𝟐﷮𝟒﷯ = 1 We need to find the height of arch at a point 1.5 m from one end. Let y be the height of arch from a point 1.5 from one end. Now, width of arch is (4 − 1.5m) = 2.5m So, point P (2.5, y) lie on ellipse it will satisfy equation of ellipse Putting x = 2.5 & y = y in (1) ﷐𝑥2﷮16﷯ + ﷐𝑦2﷮4﷯ = 1 ﷐(2.5)2﷮16﷯ + ﷐𝑦2﷮4﷯ = 1 ﷐6.25﷮16﷯ + ﷐𝑦2﷮4﷯ = 1 ﷐6.25 + 4﷐𝑦﷮2﷯﷮16﷯ = 1 6.25 + 4y2 = 16 4y2 = 16 − 6.25 4y2 = 9.75 y2 = ﷐9.75﷮4﷯ y = ﷐﷮﷐9.75﷮4﷯﷯ y = ﷐﷐﷮9.75﷯﷮2﷯ y = ﷐3.12﷮2﷯ y = 1.56 (approx) Hence, the required height is 1.56 m (approx)

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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 ask questions here.
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