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Misc 14 - In what ratio, line joining (-1, 1), (5, 7) is divided

Misc 14 - Chapter 10 Class 11 Straight Lines - Part 2
Misc 14 - Chapter 10 Class 11 Straight Lines - Part 3
Misc 14 - Chapter 10 Class 11 Straight Lines - Part 4

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Misc 14 In what ratio, the line joining (–1, 1) and (5, 7) is divided by the line x + y = 4? Let line AB is the line joining the points A(–1, 1) & B(5, 7) & let line CD be x + y = 4 Let line AB be divided by the line CD at point P Let k :1 be the ratio line AB is divided by the line CD We need to find value of k If a point divide any line joining (x1, y1) & (x2, y2) in the ratio of m1 : m2 then co-ordinate of that point is ((𝑚_2 𝑥_2 + 𝑚_1 𝑥_1)/(𝑚_1 + 𝑚_2 ),(𝑚_2 𝑦_(2 ) + 𝑚_1 𝑦_1)/(𝑚_1 + 𝑚_2 )) Point P which divide the line A(–1, 1) & B(5, 7) in k : 1 ratio is Coordinate of point P = ((5𝑘 + ( − 1))/(𝑘 + 1), (7𝑘 + 1)/(𝑘 + 1)) = ((5𝑘 − 1)/(𝑘 + 1), (7𝑘 + 1)/(𝑘 + 1)) ∴ Point P is ((5𝑘 − 1)/(𝑘 + 1), (7𝑘 + 1)/(𝑘 + 1)) Now, point P((5𝑘 − 1)/(𝑘 + 1), (7𝑘 + 1)/(𝑘 + 1)) lies on the line CD So, It will satisfy the equation of line CD So, x + y = 4 Putting x = (5𝑘 − 1)/(𝑘 + 1), y = (7𝑘 + 1)/(𝑘 + 1) ((5𝑘 − 1)/(𝑘 + 1)) + ((7𝑘 + 1)/(𝑘 + 1)) = 4 ((5𝑘 − 1) + (7𝑘 + 1))/(𝑘 + 1) = 4 5k – 1 + 7k + 1 = 4(k + 1) 5k + 7k – 1 + 1 = 4k + 4 12k + 0 = 4k + 4 12k – 4k = 4 8k = 4 k = 4/8 k = 1/2 Hence, Point P divides AB in the ratio of k : 1 = 1/2 : 1 = 2 × 1/2 : 2 × 1 = 1 : 2 Thus , required ratio is 1 : 2

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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 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.