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Ex 9.4, 18 - At (x,y) of a curve, slope of tangent is twice - Variable separation - Statement given

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  1. Chapter 9 Class 12 Differential Equations
  2. Serial order wise
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Ex 9.4, 18 At any point 𝑥 , 𝑦﷯ of a curve , the slope of the tangent is twice the slope of the line segment joining the point of contact to the point −4 , −3﷯ . Find the equation of the curve given that its passes through −2 , 1﷯ Slope of tangent to the curve = 𝑑𝑦﷮𝑑𝑥﷯ Slope of line segment joining (x, y) & (−4, −3) = 𝑦2 − 𝑦1﷮−4 + 𝑥﷯ = −3 − 𝑦﷮−4 − 𝑥﷯ = −(𝑦 + 3)﷮−(𝑥 + 4)﷯ = 𝑦 + 3﷮𝑥 + 4﷯ Given, at point (x, y). Slope of tangent is twice of line segment 𝑑𝑦﷮𝑑𝑥﷯ = 2 𝑦 + 3﷮𝑥 + 4﷯﷯ 𝑑𝑦﷮𝑦 + 3﷯ = 2 𝑑𝑥﷮𝑥 + 4﷯ Integrating both sides ﷮﷮ 𝑑𝑦﷮𝑦 + 3﷯ ﷯= 2 ﷮﷮ 𝑑𝑥﷮𝑥 + 4﷯﷯ log (y + 3) = 2 log (x + 4) + log C log (y + 3) = log (x + 4)2 + log C log (y + 3) − log (x + 4)2 = log C log 𝑦 + 3﷮ 𝑥 + 4﷯﷮2﷯﷯ = log C 𝑦 + 3﷮ 𝑥 + 4﷯﷮2﷯﷯ = C The curve passes through (−2, 1) Put x = −2 & y = 1 in (1) 1 + 3﷮ −2 + 4﷯﷮2﷯﷯ = C C = 4﷮ 2﷯﷮2﷯﷯ = 4﷮4﷯ C = 1 Put value c = 1 in equation (1) 𝑦 + 3﷮ 𝑥 + 4﷯﷮2﷯﷯ = 1 y + 3 = (x + 4)2 Hence the equation of the curve is y + 3 = (x + 4)2

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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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