Question

Assertion :ABCD is a square with vertices (0,0),(1,0),(1,1) and (0,1).P,Q,R and S are the points which divide AB,BC,CD,DA respectively in the ratio 2:1. If the origin is shifted to the centre of the square ABCD without rotation of axes, area of the square PQRS in the new system of coordinates is 59 square units. Reason: If the origin is shifted to the point P(α,β) without rotation of axes then the distance between any two given points remains unchanged in the new system of coordinates

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Solution

The correct option is **A** Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Let O be the origin and let x′Ox and y′Oy be the axis of x and y respectively. Let O be the new centre and L be two points having coordinates (h,k) and (x,y) respectively referred to as.X'OX and Y'OY as coordinates axes. Let X'OX and Y'OY be new rectangular axes

Let O be the origin and let x′Ox and y′Oy be the axis of x and y respectively. Let O be the new centre and L be two points having coordinates (h,k) and (x,y) respectively referred to as.X'OX and Y'OY as coordinates axes. Let X'OX and Y'OY be new rectangular axes

∴O′P=xandLP=YOS=xLS=yOR=hO′R=kx=OS=OR+RS=OR+O′P=h+xy=LS=LP+PS+LP+O′R=Y+K∴x=xthandy=Y+K

Thus, if (x,y) are coordinates of a point referred to old axes and (X, Y) are the coordinates o the same point referred to new axes, then

x=Xthandy=Y+Kx=2+1=3y=2+1=3

∴ The area of square PQRS in a new system

Area= b×h

=3+23+6=59

i.e, a base of a square with x-axis can be taken along with the new -coordinate =3+2=5

The height of the square with respect to x and y-axis

=3+3+3=9

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