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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 = 8Β  Β  Β  Β  Β  Β  Β  Β (B) 3x + y + 8 = 0

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

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Question 9 The equation of normal to the curve 3x2 – y2 = 8 which is parallel to the line x + 3y = 8 is (A) 3x – y = 8 (B) 3x + y + 8 = 0 (C) x + 3y Β± 8 = 0 (D) x + 3y = 0 Since, the normal to the curve is parallel to the line π‘₯+3𝑦=8 ∴ Slope of normal = Slope of line So, finding slope of normal and slope of line Finding slope of normal "3" π‘₯"2 –" 𝑦"2 = 8" Differentiating w.r.t. x 6π‘₯ βˆ’ 2𝑦 𝑑𝑦/𝑑π‘₯ = 0 2𝑦 𝑑𝑦/𝑑π‘₯ = 6π‘₯ 𝑑𝑦/𝑑π‘₯ =(6π‘₯ )/2𝑦 𝑑𝑦/𝑑π‘₯ =3π‘₯/𝑦 Slope of normal =(βˆ’1)/(𝑑𝑦/𝑑π‘₯) =(βˆ’1)/(3π‘₯/𝑦) =(βˆ’π’š)/πŸ‘π’™ Finding slope of line π‘₯+3𝑦=8 Differentiating w.r.t. x 1+3 𝑑𝑦/𝑑π‘₯ = 0 𝑑𝑦/𝑑π‘₯ = (βˆ’1)/3 Slope of line =𝑑𝑦/𝑑π‘₯ =(βˆ’πŸ)/πŸ‘ ∴ Equating (1) & (2) (βˆ’π’š)/πŸ‘π’™ = (βˆ’πŸ)/πŸ‘ βˆ’3𝑦=βˆ’3π‘₯ π’š=𝒙 Now, to find equation of normal, we need a point So, Putting π’š=𝒙 in the curve 3𝑦^2βˆ’π‘₯^2=8 3π‘₯^2βˆ’π‘₯^2=8 2π‘₯^2 = 8 π‘₯^2 = 8/2 π‘₯^2 = 4 𝒙=±𝟐 For y-coordinates, putting value of π‘₯ in y=π‘₯ Finding equation of normal Equation of line at (π‘₯_1, 𝑦_1) & having slope m is (π‘¦βˆ’π‘¦_1 ) = m (π‘₯βˆ’π‘₯_1 ) For 𝒙 = 2 𝑦=π‘₯ 𝑦=2 So, the point is (2, 2) For 𝒙 = βˆ’2 𝑦=π‘₯ 𝑦=βˆ’2 So, the point is (βˆ’2, βˆ’2) Equation of normal at (2, 2) & Slope (βˆ’πŸ)/πŸ‘ (π‘¦βˆ’2) = (βˆ’1)/3 (π‘₯βˆ’2) 3 (π‘¦βˆ’2) = βˆ’1 (π‘₯βˆ’2) 3π‘¦βˆ’6=βˆ’π‘₯+2 3𝑦+π‘₯=6+2 3𝑦+π‘₯=8 πŸ‘π’š+π’™βˆ’πŸ–=𝟎 Equation of normal at (βˆ’2, βˆ’2) & Slope (βˆ’πŸ)/πŸ‘ (𝑦+2) = (βˆ’1)/3 (π‘₯+2) 3 (𝑦+2) = βˆ’1 (π‘₯+2) 3𝑦+6=βˆ’π‘₯βˆ’2 3𝑦+π‘₯=βˆ’6βˆ’2 3𝑦+π‘₯=βˆ’8 πŸ‘π’š+𝒙+πŸ–=𝟎 Hence, the required equation of normal is 3y + 𝒙 Β± 8 = 0 So, the correct answer is (C)

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.