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Ex 6.3, 20 - Find equation of normal at (am2, am3) for ay2 = x3 - Finding equation of tangent/normal when point and curve is given

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  1. Chapter 6 Class 12 Application of Derivatives
  2. Serial order wise
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Ex 6.3,20 Find the equation of the normal at the point (𝑎﷐𝑚﷮2﷯,𝑎﷐𝑚﷮3﷯) for the curve 𝑎﷐𝑦﷮2﷯=﷐𝑥﷮3﷯ We know that Slope of tangent is ﷐𝑑𝑦﷮𝑑𝑥﷯ Given 𝑎﷐𝑦﷮2﷯=﷐𝑥﷮3﷯ Differentiating w.r.t.𝑥 ﷐𝑑﷐𝑎﷐𝑦﷮2﷯﷯﷮𝑑𝑥﷯=﷐𝑑﷐﷐𝑥﷮3﷯﷯﷮𝑑𝑥﷯ 𝑎﷐𝑑﷐﷐𝑦﷮2﷯﷯﷮𝑑𝑥﷯=﷐𝑑﷐﷐𝑥﷮3﷯﷯﷮𝑑𝑥﷯ 𝑎 . ﷐𝑑﷐﷐𝑦﷮2﷯﷯﷮𝑑𝑥﷯× ﷐𝑑𝑦﷮𝑑𝑦﷯=3﷐𝑥﷮2﷯ 𝑎 .2𝑦 ×﷐𝑑𝑦﷮𝑑𝑥﷯=3﷐𝑥﷮2﷯ ﷐𝑑𝑦﷮𝑑𝑥﷯=﷐3﷐𝑥﷮2﷯﷮2𝑎𝑦﷯ Slope of tangent at ﷐𝑎﷐𝑚﷮2﷯,𝑎﷐𝑚﷮3﷯﷯ is ﷐﷐𝑑𝑦﷮𝑑𝑥﷯│﷮﷐𝑎﷐𝑚﷮2﷯,𝑎﷐𝑚﷮3﷯﷯﷯=﷐3﷐﷐𝑎﷐𝑚﷮2﷯﷯﷮2﷯﷮2𝑎﷐𝑎﷐𝑚﷮3﷯﷯﷯=﷐3﷐𝑎﷮2﷯﷐𝑚﷮4﷯﷮2﷐𝑎﷮2﷯﷐𝑚﷮3﷯﷯=﷐3﷮2﷯𝑚 We know that Slope of tangent × Slope of Normal =−1 ﷐3𝑚﷮2﷯ × Slope of Normal =−1 Slope of Normal =﷐−1﷮﷐3𝑚﷮2﷯﷯ Slope of Normal =﷐−2﷮3𝑚﷯ Finding equation of normal Equation of Normal at ﷐𝑎﷐𝑚﷮2﷯, 𝑎﷐𝑚﷮3﷯﷯ & having Slope ﷐−2﷮3𝑚﷯ is ﷐𝑦−𝑎﷐𝑚﷮3﷯﷯=﷐−2﷮3𝑚﷯﷐𝑥−𝑎﷐𝑚﷮2﷯﷯ 3𝑚﷐𝑦−𝑎﷐𝑚﷮3﷯﷯=−2﷐𝑥−𝑎﷐𝑚﷮2﷯﷯ 3𝑚𝑦−3 𝑎﷐𝑚﷮4﷯=−2𝑥+2𝑎﷐𝑚﷮2﷯ 2𝑥+3𝑚𝑦−3𝑎﷐𝑚﷮4﷯−2𝑎﷐𝑚﷮2﷯=0 2𝑥+3𝑚𝑦−𝑎﷐𝑚﷮2﷯﷐3﷐𝑚﷮2﷯+2﷯=0 Required Equation of Normal is : 𝟐𝒙+𝟑𝒎𝒚−𝒂﷐𝒎﷮𝟐﷯﷐𝟑﷐𝒎﷮𝟐﷯+𝟐﷯=𝟎

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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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