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  • Let the algebraic sum of the perpendicular distances from the points <img alt="\mathrm{A(2,0), B(0,2), C(1,1)}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7BA%282%2C0%29%2C%20B%

Let the algebraic sum of the perpendicular distances from the points \mathrm{A(2,0), B(0,2), C(1,1)} to a variable straight line be zero. Then, the line passes through a fixed point
 

Option: 1

(1,0)

 


Option: 2

(1,1)
 


Option: 3

(1,2)
 


Option: 4

(2,2)


Answers (1)

best_answer

Let the line be \mathrm{l x+m y+n=0} and perpendicular distances be \mathrm{p_1, p_2, p_3} from \mathrm{A, B, C} respectively.

\mathrm{ \therefore p_1+p_2+p_3=\frac{2 l+n}{\sqrt{l^2+m^2}}+\frac{2 m+n}{\sqrt{l^2+m^2}}+\frac{l+m+n}{\sqrt{l^2+m^2}}=0 }

\mathrm{ \Rightarrow 3 l+3 m+3 n=0 }

\mathrm{ \Rightarrow l+m+n=0}

Or (1,1) lies on the line, \mathrm{l x+m y+n=0}.

Hence option 2 is correct

Posted by

manish

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