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Question

A point moves such that the sum of the square of its distance from the sides of a sqaure of side unity is equal to $$9$$. The locus of the point is a circle such that


A
Center of the circle coincides with that of square 
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B
Center of the circle is (12,12)
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C
Radius of the circle is 2
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D
All the above are true
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Solution

The correct option is D All the above are true
Let the sides of the square be $$y=0,y=1,x=0$$ and $$x=1.$$
Let the moving point be $$(x,y).$$
Then,$${ y }^{ 2 }+{ \left( y-1 \right)  }^{ 2 }+{ x }^{ 2 }+{ \left( x-1 \right)  }^{ 2 }=9$$ is the equation of the locus.
$$\Rightarrow 2{ x }^{ 2 }+2{ y }^{ 2 }-2x-2y-7=0,$$ 
which represents a circle having centre $$\displaystyle \left( \frac { 1 }{ 2 } ,\frac { 1 }{ 2 }  \right) $$ (which is also the centre of the square) and radius $$2.$$  

Mathematics

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