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

Transcript

Ex 6.3, 25 Find the equation of the tangent to the curve โˆš(3๐‘ฅโˆ’2) which is parallel to the line 4x โˆ’ 2y + 5 = 0 . Let (โ„Ž , ๐‘˜) be the point on Curve from tangent to be taken We know that Equation of tangent is ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ ๐‘ฆ=โˆš(3๐‘ฅ โˆ’2) Differentiating w.r.t.๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ=(๐‘‘(3๐‘ฅ โˆ’2)^(1/2))/๐‘‘๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ=3/(2โˆš(3๐‘ฅ โˆ’2)) Slope of tangent at (โ„Ž , ๐‘˜) is ใ€–๐‘‘๐‘ฆ/๐‘‘๐‘ฅโ”‚ใ€—_((โ„Ž , ๐‘˜) )=3/(2โˆš(3โ„Ž โˆ’ 2)) Given tangent is parallel to the line 4๐‘ฅโˆ’2๐‘ฆ+5 So , Slope of tangent = Slope of 4๐‘ฅโˆ’2๐‘ฅ+5 Now, Given line is 4๐‘ฅโˆ’2๐‘ฆ+5=0 โˆ’2๐‘ฆ=โˆ’4๐‘ฅโˆ’5 2๐‘ฆ=4๐‘ฅ+5 ๐‘ฆ=(4๐‘ฅ + 5)/2 ๐‘ฆ=2๐‘ฅ+5/2 The above Equation is of the form ๐‘ฆ=๐‘š๐‘ฅ+๐‘ where m is Slope of line โˆด Slope of line is 2 Now, Slope of tangent at (โ„Ž , ๐‘˜)= Slope of line 4๐‘ฅโˆ’3๐‘ฆ+5=0 3/(2โˆš(3โ„Ž โˆ’2))=2 3=2 ร—2โˆš(3โ„Ž โˆ’2) 3=4โˆš(3โ„Ž โˆ’2) Squaring Both Sides (3)^2=(4โˆš(3โ„Ž โˆ’2))^2 9=(4)^2 (โˆš(3โ„Ž โˆ’2))^2 9=16(3โ„Ž โˆ’2) 9/16=3โ„Ž โˆ’2 3โ„Ž โˆ’2=9/16 3โ„Ž=9/16+2 3โ„Ž=(9 + 32)/16 3โ„Ž=41/16 โ„Ž=41/(16 ร— 3) โ„Ž=41/48 Now, ๐‘ฆ=โˆš(3๐‘ฅ โˆ’2) Since Point (โ„Ž , ๐‘˜) is on the Curve Point (โ„Ž , ๐‘˜) Satisfies the Equation of Curve Putting ๐‘ฅ=โ„Ž , ๐‘ฆ=๐‘˜ ๐‘˜=โˆš(3โ„Ž โˆ’2) Finding k when โ„Ž=41/48 ๐‘˜=โˆš(3 ร—41/48โˆ’2) =โˆš(41/16โˆ’2)=โˆš((41 โˆ’32)/16)=โˆš(9/16)=3/4 Hence the point is (h, k) = (41/48 , 3/4) We know that Equation of line at (๐‘ฅ1 , ๐‘ฆ1)& having Slope m is ๐‘ฆโˆ’๐‘ฆ1=๐‘š(๐‘ฅโˆ’๐‘ฅ1) Equation of tangent at (41/48 , 3/4) & having Slope 2 is (๐‘ฆโˆ’3/4)=2(๐‘ฅโˆ’41/48) (4๐‘ฆ โˆ’ 3)/4=2((48๐‘ฅ โˆ’ 41)/48) (4๐‘ฆ โˆ’ 3)/4=(48๐‘ฅ โˆ’ 41)/24 24(4๐‘ฆ โˆ’ 3)/4=48๐‘ฅโˆ’41 6(4๐‘ฆโˆ’3)=48๐‘ฅโˆ’41 24๐‘ฆโˆ’18=48๐‘ฅโˆ’41 48๐‘ฅโˆ’41โˆ’24๐‘ฆ+18=0 48๐‘ฅโˆ’24๐‘ฆโˆ’23=0 48๐‘ฅโˆ’24๐‘ฆ=23 Hence Required Equation of tangent is ๐Ÿ’๐Ÿ–๐’™โˆ’๐Ÿ๐Ÿ’๐’š=๐Ÿ๐Ÿ‘

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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.