Last updated at April 19, 2021 by Teachoo

Transcript

Example 35 Find the shortest distance of the point (0, c) from the parabola ๐ฆ=๐ฅ2, where 0 โค c โค 5. Let (โ ,๐) be any point on parabola ๐ฆ=๐ฅ2 Let D be required Distance between (โ , ๐) & (0 , ๐) D = โ((0โโ)^2+(๐ โ๐)^2 ) D = โ((โโ)^2+(๐ โ๐)^2 ) D = โ(๐^๐+(๐ โ๐)^๐ ) Distance between two (๐ฅ1,๐ฆ1) & (๐ฅ2 , ๐ฆ2) point is ๐= โ((๐ฅ2โ๐ฆ1)^2+(๐ฅ2 โ๐ฆ1)^2 ) Also, Since point (โ , ๐) is on the parabola ๐ฆ=๐ฅ2 (๐ , ๐) will satisfy the equation of parabola Putting ๐ฅ=โ , ๐ฆ=๐ in equation ๐=๐^๐ Putting value of ๐=โ^2 D = โ(โ^2+(๐ โ๐)^2 ) D = โ(๐+(๐โ๐)^๐ ) We need to minimize D, but D has a square root Which will be difficult to differentiate Let Z = D2 Z = ๐+(๐โ๐)^2 Since D is positive, D is minimum if D2 is minimum So, we minimize Z = D2 Differentiating Z Z =๐+(๐โ๐)^2 Differentiating w.r.t. k Zโ = ๐(๐ + (๐ โ ๐)^2 )/๐๐ Zโ = 1 + 2 (c โ k) ร (c โ k)โ Zโ = 1 + 2 (c โ k) ร (0 โ 1) Zโ = 1 โ 2 (c โ k) Zโ = 1 โ 2c โ 2k Putting Zโ = 0 1 โ 2c โ 2k = 0 2k = 2c โ 1 k = (๐๐ โ ๐)/๐ Now, checking sign of ๐^โฒโฒ " " ๐๐/๐๐=4๐โ2๐ Differentiating again w.r.t k (๐^2 ๐)/(๐โ^2 ) = 4 โ0 (๐ ^๐ ๐)/(๐ ๐^๐ ) = ๐ Since ๐^โฒโฒ > 0 for k = (2๐ โ 1)/2 โด Z is minimum when k = (2๐ โ 1)/2 Thus, D is Minimum at ๐=(๐๐ โ ๐)/๐ Finding Minimum value of D D = โ(๐+(๐โ๐)^2 ) Putting ๐=(2๐ โ 1)/2 D = โ(((2๐ โ 1)/2)+(๐โ((2๐ โ 1)/2))^2 ) D = โ(((2๐ โ 1)/2)+((2๐ โ 2๐ โ 1)/2)^2 ) D = โ(((2๐ โ 1)/2)+((โ1)/2)^2 ) D = โ(((2๐ โ 1)/2)+1/4) D = โ(๐โ1/2+1/4) D = โ(๐โ1/4) D = โ(4๐ โ 1)/2 Hence, shortest distance is โ(๐๐ โ ๐)/๐

Examples

Example 1
Deleted for CBSE Board 2022 Exams

Example 2 Deleted for CBSE Board 2022 Exams

Example 3 Deleted for CBSE Board 2022 Exams

Example 4 Important Deleted for CBSE Board 2022 Exams

Example 5 Deleted for CBSE Board 2022 Exams

Example 6 Deleted for CBSE Board 2022 Exams

Example 7

Example 8 Important

Example 9 Important

Example 10

Example 11 Important

Example 12

Example 13 Important

Example 14

Example 15

Example 16

Example 17 Important

Example 18

Example 19

Example 20

Example 21 Deleted for CBSE Board 2022 Exams

Example 22 Deleted for CBSE Board 2022 Exams

Example 23 Deleted for CBSE Board 2022 Exams

Example 24 Deleted for CBSE Board 2022 Exams

Example 25 Deleted for CBSE Board 2022 Exams

Example 26

Example 27

Example 28 Important

Example 29

Example 30 Important

Example 31

Example 32 Important

Example 33 Important

Example 34

Example 35 Important You are here

Example 36

Example 37 Important

Example 38 Important

Example 39

Example 40 Important

Example 41 Important

Example 42 Important Deleted for CBSE Board 2022 Exams

Example 43 Important Deleted for CBSE Board 2022 Exams

Example 44 Important Deleted for CBSE Board 2022 Exams

Example 45 Important Deleted for CBSE Board 2022 Exams

Example 46 Important

Example 47 Important

Example 48 Important

Example 49 Deleted for CBSE Board 2022 Exams

Example 50 Important

Example 51

About the Author

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.