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  • The point 4 1 undergoes the following two successive transformations i Reflection about the line y equal t0 x ii Translation through a distance 2 units along the positive xaxis Then the final coordinates of the point are A 4 3 B 3 4 C 1 4 D 7/2 7/2

The point (4, 1) undergoes the following two successive transformations:
(i) Reflection about the line y = x
(ii) Translation through a distance of 2 units along the positive x-axis. Then the final coordinates of the point are
A. (4, 3)
B. (3, 4)
C. (1, 4)
D. \frac{7}{2}, \frac{7}{2}

Answers (1)

Let Q(x,y) be the reflection of P(4,1) about  the line y=x, then

 midpoint of PQ  \left ( \frac{4+x}{2},\frac{1+y}{2} \right )

which lies on y=x  \frac{4+x}{2}=\frac{1+y}{2}

   4+x=1+y   

  x-y+3=0……(i)  

  Now, we find the slope of given equation y=x  

 Since this equation is in y=mx+b form 

So the slope=m=1 

 Slope of PQ=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{y-1}{x-4}

 Since, PQ is perpendicular to y=x

 And when two lines are perpendicular, then m1m2=-1 

1*\left ( \frac{y-1}{x-4} \right )=-1

    y-1=-(x-4)   

   x+y-5=0…..(ii)  

On adding equation i and ii, we get  x-y+3+x+y-5=0  

  2x-2=0 

  x-1=0 

 x=1  

 Putting the value of x=1 in equation i we get  1-y+3=0 -y+4=0  y=4  

 It is given that translation through a distance of 2 units along the positive x axis takes place

   The point after translation is (1+2,4)=(3 ,4)

Hence, the correct option is (b)

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