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Solution


Q.

The vertices of a triangle are A(–1, 3), B(–2, 2) and C(3, –1). A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is            [2024]
1 -x+y-(2-2)=0  
2 x-y-(2+2)=0  
3 x+y+(2-2)=0  
4 x+y-(2-2)=0  

Ans.

(4)

Equation of AC : y - 3 = -44 (x + 1)

⇒   x + y = 2

Equation AB : y - 2 = 11 (x + 2)

⇒   x - y + 4 = 0

Equation BC : y + 1 = -35 (x - 3)

⇒   3x + 5y = 4

Distance of AC from origin O(0, 0) = |-2|2 = 2

Distance of AB from origin O(0, 0) = |4|2 = 22

Distance of BC from origin O(0, 0) = |-4|9 + 25 = 434

When each side of ∆ABC shifted by one unit inwards distance of new sides from origin are 2 - 1, 22 - 1, 434 + 1 respectively.

Clearly new position of AC is nearest to the origin.

Equation of AC when shifted by one unit inwareds can be chosen as x + y = c

Thus, we have |c - 2|2 = 1  ⇒  c = 2 - 2

∴  Required equation is x + y - 2 + 2 = 0.