On shifting the origin to the point \(\left(\frac{1}{2}, -\frac{1}{3}\right)\) and keeping the axes parallel, the new coordinates of the point \(\left(-\frac{1}{5}, \frac{1}{3}\right)\) will be:
(1) \(\left(\frac{7}{10}, -\frac{2}{3}\right)\)
(2) \(\left(-\frac{1}{10}, \frac{2}{3}\right)\)
(3) \(\left(-\frac{7}{10}, \frac{2}{3}\right)\)
(4) \(\left(-\frac{7}{10}, 0\right)\)
Answer (1)
To solve this problem, we need to find the new coordinates of the point ( − 5 1 , 3 1 ) after shifting the origin to the point ( 2 1 , − 3 1 ) , with the axes remaining parallel.
Step-by-Step Solution
Understanding the Shift:
When you shift the origin to a new point ( h , k ) , any point ( x , y ) in the original coordinate system translates to ( x ′ , y ′ ) in the new system.
The transformation equations are: x ′ = x − h y ′ = y − k
Applying the Shift:
In this problem, h = 2 1 and k = − 3 1 .
Apply these values to the given point ( − 5 1 , 3 1 ) .
Calculate x ′ : [
x' = -\frac{1}{5} - \frac{1}{2} ] To subtract these two fractions, find a common denominator: x ′ = − 10 2 − 10 5 = − 10 7
Calculate y ′ : y ′ = 3 1 − ( − 3 1 ) This is equivalent to adding 3 1 : y ′ = 3 1 + 3 1 = 3 2
Result of the Shift:
The new coordinates of the point are ( − 10 7 , 3 2 ) .
Therefore, the correct choice is option (3): ( − 10 7 , 3 2 ) .
