The equation of normal to the curve 3x 2 – y 2 = 8 which is parallel to the line x + 3y = 8 is

(A) 3x – y = 8               (B) 3x + y + 8 = 0

(C) x + 3y ± 8 = 0         (D) x + 3y = 0

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  1. Chapter 6 Class 12 Application of Derivatives (Term 1)
  2. Serial order wise

Transcript

Question 9 The equation of normal to the curve 3x2 โ€“ y2 = 8 which is parallel to the line x + 3y = 8 is (A) 3x โ€“ y = 8 (B) 3x + y + 8 = 0 (C) x + 3y ยฑ 8 = 0 (D) x + 3y = 0 Since, the normal to the curve is parallel to the line ๐‘ฅ+3๐‘ฆ=8 โˆด Slope of normal = Slope of line So, finding slope of normal and slope of line Finding slope of normal "3" ๐‘ฅ"2 โ€“" ๐‘ฆ"2 = 8" Differentiating w.r.t. x 6๐‘ฅ โˆ’ 2๐‘ฆ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 0 2๐‘ฆ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 6๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ =(6๐‘ฅ )/2๐‘ฆ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ =3๐‘ฅ/๐‘ฆ Slope of normal =(โˆ’1)/(๐‘‘๐‘ฆ/๐‘‘๐‘ฅ) =(โˆ’1)/(3๐‘ฅ/๐‘ฆ) =(โˆ’๐’š)/๐Ÿ‘๐’™ Finding slope of line ๐‘ฅ+3๐‘ฆ=8 Differentiating w.r.t. x 1+3 ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 0 ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = (โˆ’1)/3 Slope of line =๐‘‘๐‘ฆ/๐‘‘๐‘ฅ =(โˆ’๐Ÿ)/๐Ÿ‘ โˆด Equating (1) & (2) (โˆ’๐’š)/๐Ÿ‘๐’™ = (โˆ’๐Ÿ)/๐Ÿ‘ โˆ’3๐‘ฆ=โˆ’3๐‘ฅ ๐’š=๐’™ Now, to find equation of normal, we need a point So, Putting ๐’š=๐’™ in the curve 3๐‘ฆ^2โˆ’๐‘ฅ^2=8 3๐‘ฅ^2โˆ’๐‘ฅ^2=8 2๐‘ฅ^2 = 8 ๐‘ฅ^2 = 8/2 ๐‘ฅ^2 = 4 ๐’™=ยฑ๐Ÿ For y-coordinates, putting value of ๐‘ฅ in y=๐‘ฅ Finding equation of normal Equation of line at (๐‘ฅ_1, ๐‘ฆ_1) & having slope m is (๐‘ฆโˆ’๐‘ฆ_1 ) = m (๐‘ฅโˆ’๐‘ฅ_1 ) For ๐’™ = 2 ๐‘ฆ=๐‘ฅ ๐‘ฆ=2 So, the point is (2, 2) For ๐’™ = โˆ’2 ๐‘ฆ=๐‘ฅ ๐‘ฆ=โˆ’2 So, the point is (โˆ’2, โˆ’2) Equation of normal at (2, 2) & Slope (โˆ’๐Ÿ)/๐Ÿ‘ (๐‘ฆโˆ’2) = (โˆ’1)/3 (๐‘ฅโˆ’2) 3 (๐‘ฆโˆ’2) = โˆ’1 (๐‘ฅโˆ’2) 3๐‘ฆโˆ’6=โˆ’๐‘ฅ+2 3๐‘ฆ+๐‘ฅ=6+2 3๐‘ฆ+๐‘ฅ=8 ๐Ÿ‘๐’š+๐’™โˆ’๐Ÿ–=๐ŸŽ Equation of normal at (โˆ’2, โˆ’2) & Slope (โˆ’๐Ÿ)/๐Ÿ‘ (๐‘ฆ+2) = (โˆ’1)/3 (๐‘ฅ+2) 3 (๐‘ฆ+2) = โˆ’1 (๐‘ฅ+2) 3๐‘ฆ+6=โˆ’๐‘ฅโˆ’2 3๐‘ฆ+๐‘ฅ=โˆ’6โˆ’2 3๐‘ฆ+๐‘ฅ=โˆ’8 ๐Ÿ‘๐’š+๐’™+๐Ÿ–=๐ŸŽ Hence, the required equation of normal is 3y + ๐’™ ยฑ 8 = 0 So, the correct answer is (C)

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.