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Question

ABCD is a rhombus. The slope of 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). If the area of rhombus is 102 sq. units, then which of the following is (are) CORRECT?

A
Length of smaller diagonal is 4
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B
Length of larger diagonal is 42
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C
One of the vertex is (2,1)
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D
Perimeter of rhombus is 257
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Solution

The correct option is D Perimeter of rhombus is 257
The diagonal AC is the member of family of lines (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 is 1,
therefore equation of AC is, 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=12(d1×d2)=102
d1=5
Side of rhombus=d214+d224=572
Perimeter=257

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