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A diagonal of rhombus ABCD is a member of both the family lines (x+y-1)+k1(2x+3y-2)=0 and (x-y+2)+k2(2x-3y+5) where k1,k2 are Real Numb and one of the vertex of the rhombus is (3,2). If the area of the rhoombus is 12√5 sq.units then find the length of semi longer diagonal of the rhombus

A diagonal of rhombus ABCD is a member of both the family lines (x+y-1)+k1(2x+3y-2)=0 and (x-y+2)+k2(2x-3y+5) where k1,k2 are Real Numb and one of the vertex of the rhombus is (3,2). If the area of the rhoombus is 12√5 sq.units then find the length of semi longer diagonal of the rhombus

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1 Answers

Satyajit Samal
19 Points
6 years ago
One diagonal is a member of both the family of lines (x+y-1)+k1(2x+3y-2)=0 and (x-y+2)+k2(2x-3y+5)
Hence it must pass through the point of intersection of x+y-1= 0 and 2x+3y-2 = 0 .
Also it should also pass through the point of intersection of x-y+2 = 0 and 2x-3y+5 = 0 .
Solving both pairs of equations, we get points as (1,0) and ( -1, 1) . 
Hence equation of one diagonal of the rhombus which passes through the above two points is 
y-0 = -1/ 2 ( x – 1 ) ; 2y = –x + 1 …..........(1)
One of the vertex has coordinates (3,2) , which does not satisfy the above equation. So, it must lie on the other diagonal which is perpendicular to the diagonal 2y = –x +1 . 
Hence, it’s equation can be of form y = 2x + c. Since it passes through (3, 2) we get the equation as y = 2x-4 ….......... (2)
Now, solve (1) and (2) to find the point of intersection. We get the coordinates as ( 9/5 , -2/5) .
Hence the other vertex on 2nd Diagonal is the point with coordinates x = 3 / 5 and y = -14 / 5  ( using mid point formula) 
distance between (3,2) and ( 3/5 , –14/5) is 12 / sqrt(5) = d2
Area = ½ d1d2= 12 sqrt(5) , we get d1 = 10 .
Hence, length of semi-longer diagnonal is 5. 
 
 
 

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