Question

- What are the new co-ordinates of a point \[\left( {4,5} \right)\], when the origin is shifted to the point \[\left( {1, - 2} \right)\]?
A. \[\left( {5,3} \right)\]
B. \[\left( {3,5} \right)\]
C. \[\left( {3,7} \right)\]
D. None

Answer 1

- Hint: After shifting of origin, the co-ordinate of each point changes according to the new place of origin. Use the relations between the new co-ordinates and the old co-ordinates of a point and find the required point.

Formula used: If origin is shifted to the point \[\left( {h,k} \right)\], the old co-ordinates of a point be \[\left( {x,y} \right)\] and the new co-ordinates of that point be \[\left( {x',y'} \right)\], then \[x = x' + h\] and \[y = y' + k\].

Complete step-by-step solution:
 Here \[x = 4,y = 5,h = 1,k = - 2\]
Now, use the relations to find the values of \[x'\] and \[y'\].
Using \[x = x' + h\], we get
\[4 = x' + 1\]
\[\begin{array}{l} \Rightarrow x' = 4 - 1\\ \Rightarrow x' = 3\end{array}\]
Using \[y = y' + k\], we get
\[5 = y' - 2\]
\[\begin{array}{l} \Rightarrow y' = 5 + 2\\ \Rightarrow y' = 7\end{array}\]
Finally, we get \[x' = 3\] and \[y' = 7\]
Put these values in the co-ordinate \[\left( {x',y'} \right)\].
So, the new co-ordinate is \[\left( {3,7} \right)\].
Hence, option C is correct.

Additional information:
The origin of the axes is shifted at \[\left(h,k\right)\]. This means the x-axis is shifted h units and the y-axis is shifted k units with respect to the original position.
The intersection point of the new position of the axes is the new origin of the system.

Note: You should be careful about the relations. Many students can’t remember the relations properly. Usually, they write \[x\] at the place of \[x'\] and write \[y\] at the place of \[y'\] and also they mess up with the signs.