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 $\frac{5}{9}$ square units. Reason: If the origin is shifted to the point $P(\alpha ,\beta )$ without rotation of axes then the distance between any two given points remains unchanged in the new system of coordinates

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

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 $x^{\prime }Ox$ and $y^{\prime }Oy$ 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

$\therefore O^{\prime }P=xandLP=YOS=xLS=yOR=h{O}^{\prime }R=kx=OS=OR+RS=OR+O^{\prime }P=h+xy=LS=LP+PS+LP+O^{\prime }R=Y+K\therefore x=xthandy=Y+K$

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

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

$\therefore$ The area of square PQRS in a new system

Area= $b\times$h

$=\frac{3+2}{3+6}=\frac{5}{9}$

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

$=3+3+3=9$