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Question 1 Important Deleted for CBSE Board 2024 Exams

Last updated at April 16, 2024 by Teachoo

Misc 13 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