St. Lines

Ans:

One diagonal is a member of both the family of lines (x+y1)+k1(2x+3y2)=0 and (xy+2)+k2(2x3y+5)
Hence it must pass through the point of intersection of x+y1= 0 and 2x+3y2 = 0 .
Also it should also pass through the point of intersection of xy+2 = 0 and 2x3y+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
y0 = 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 = 2x4 ….......... (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 semilonger diagnonal is 5.