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  •  What is represented by the locus<img alt="\mathrm{ (x-y+c)^2+(x+y-c)^2=0 \text { ? }}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%20%28x-y&plus;c%29%5E2&plus;%2

 What is represented by the locus\mathrm{ (x-y+c)^2+(x+y-c)^2=0 \text { ? }}

Option: 1

Pair of coincident lines.


Option: 2

Pair of intersecting lines.


Option: 3

Pair of imaginary lines passing through \mathrm{(0, c)}


Option: 4

None of these.


Answers (1)

best_answer

We know that the sum of the squares of two real quantities cannot be zero unless each of the squares is separately zero. The only real points that satisfy the equation (1) therefore satisfy both of the equations

\mathrm{x-y+c=0 } and  \mathrm{ x+y-c=0}
But the only solution of these two equation \mathrm{ x=0}, and \mathrm{ y=c}.
The only real point represented by equation (1) is therefore\mathrm{ (0, c)}.
The same result may be obtained in a different manner.
The equation (1) gives \mathrm{(x-y-c)^2=-(x+y-c)^2},

\mathrm{\text { i.e. } x-y+c= \pm \sqrt{-1}(x+y-c)}

It therefore represents the two imaginary straight lines

\mathrm{x^{(1-\sqrt{-1})}-y^{(1+\sqrt{-1})}+c(1+\sqrt{-1})=0} 

and, \mathrm{x^{(1+\sqrt{-1})}-y^{(1-\sqrt{-1})}+c(1-\sqrt{-1})=0}

Each of these two straight lines passes through the real points \mathrm{(0, c)}. We may therefore say that (1) represents two imaginary straight lines passing through the point \mathrm{(0, \mathrm{c})}.

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jitender.kumar

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