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Question

In a rhombus ABCD, the slope of diagonal AC is 1 and is among the family of lines (x+2y5)+λ(3x+y5)=0, where λR. One of the vertex of the rhombus is (2,3) and the area of rhombus is 102 sq. units, then which of the following is/are correct?

A
Length of smaller diagonal is 5
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B
Length of larger diagonal is 62
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C
One of the vertex is (2,1)
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D
Perimeter of rhombus is 257 units
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Solution

The correct options are
A Length of smaller diagonal is 5
C One of the vertex is (2,1)
D Perimeter of rhombus is 257 units
The diagonal AC is the member of family of line (x+2y5)+λ(3x+y5)=0, so it will pass through the intersection point of lines x+2y5=0 and 3x+y5=0 i.e. (1,2).

Since, slope of AC=1, therefore equation of AC:xy+1=0
The point (2,3) does not lie on xy+1=0, so it will lie on another diagonal BD.
Since, diagonals of rhombus are perpendicular to each other, therefore equation of BD is, x+y1=0


Point of intersection of AC and BD is (0,1)
B:(x,y)=(2,1)
BD=42=d2

Area of rhombus =102
12(d1×d2)=102
d1=5

Side of rhombus =d214+d224=572
Perimeter =257

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