In a rhombus ABCD, the slope of diagonal AC is 1 and is among the family of lines (x+2y−5)+λ(3x+y−5)=0, where λ∈R. One of the vertex of the rhombus is (−2,3) and the area of rhombus is 10√2 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 6√2
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C
One of the vertex is (2,−1)
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D
Perimeter of rhombus is 2√57 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 2√57 units The diagonal AC is the member of family of line (x+2y−5)+λ(3x+y−5)=0, so it will pass through the intersection point of lines x+2y−5=0 and 3x+y−5=0 i.e. (1,2).
Since, slope of AC=1, therefore equation of AC:x−y+1=0 The point (−2,3) does not lie on x−y+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+y−1=0
Point of intersection of AC and BD is (0,1) ⇒B:(x,y)=(2,−1) BD=4√2=d2
Area of rhombus =10√2 ⇒12(d1×d2)=10√2 ⇒d1=5
Side of rhombus =√d214+d224=√572 ∴ Perimeter =2√57