Science > Mathematics > Coordinate Geometry > Locus > Shift of Origin
Type – I: To Find New Coordinates After Shift of Origin
Example – 01:
If the origin is shifted to a point (1, 2), axes remaining parallel, find the new coordinates of the point (2, -3)
Solution:
New origin (1, 2) ≡ (h, k)
Let (x, y) ≡ (2, -3) be the old coordinates and (X, Y) be the new coordinates
We have X = x – h and Y = y – k
∴ X = 2 – 1 = 1 and Y = -3 -2 = -5
Ans: The new coordinates of point (2, -3) are (1, -5)
Example – 02:
If the origin is shifted to a point (1, 2), axes remaining parallel, find the new coordinates of the point (3, -4)
Solution:
New origin (1, 2) ≡ (h, k)
Let (x, y) ≡ (3, -4) be the old coordinates and (X, Y) be the new coordinates
We have X = x – h and Y = y – k
∴ X = 3 – 1 = 2 and Y = -4 -2 = -6
Ans: The new coordinates of point (3, -4) are (2, -6)
Example – 03:
If the origin is shifted to a point (2, -3), axes remaining parallel, find the new coordinates of the point (5, 2)
Solution:
New origin (2, -3) ≡ (h, k)
Let (x, y) ≡ (5, 2) be the old coordinates and (X, Y) be the new coordinates
We have X = x – h and Y = y – k
∴ X = 5 – 2 = 3 and Y = 2 + 3 = 5
Ans: The new coordinates of point (5, 2) are (3, 5)
Example – 04:
If the origin is shifted to a point (2, -3), axes remaining parallel, find the new coordinates of the point (2, -7)
Solution:
New origin (2, -3) ≡ (h, k)
Let (x, y) ≡ (2, -7) be the old coordinates and (X, Y) be the new coordinates
We have X = x – h and Y = y – k
∴ X = 2 – 2 = 0 and Y = -7 + 3 = -4
Ans: The new coordinates of point (2, -7) are (0, -4)
Example – 05:
If the origin is shifted to a point (2, -3), axes remaining parallel, find the new coordinates of the point (3, -4)
Solution:
New origin (2, -3) ≡ (h, k)
Let (x, y) ≡ (3, -4) be the old coordinates and (X, Y) be the new coordinates
We have X = x – h and Y = y – k
X = 3 – 2 = 1 and Y = -4 + 3 = -1
Ans: The new coordinates of point (3, -4) are (1, -1)
Example – 06:
If the origin is shifted to a point (2, -3), axes remaining parallel, find the new coordinates of the point (-10, 3)
Solution:
New origin (2, -3) ≡ (h, k)
Let (x, y) ≡ (-10, 3) be the old coordinates and (X, Y) be the new coordinates
We have X = x – h and Y = y – k
∴ X = -10 – 2 = -12 and Y = 3 + 3 = 6
Ans: The new coordinates of point (-10, 3) are (-12, 6)
Example – 07:
If the origin is shifted to a point (2, -3), axes remaining parallel, find the new coordinates of the point (2, -3)
Solution:
New origin (2, -3) ≡ (h, k)Let (x, y) ≡ (2, -3) be the old coordinates and (X, Y) be the new coordinates
We have X = x – h and Y = y – k
∴ X = 2 – 2 = 0 and Y = -3 + 3 = 0
Ans: The new coordinates of point (2, -3) are (0, 0)
Type – II: To Find Old Coordinates After Shift of Origin
Example – 08:
If the origin is shifted to a point (-3, 2), axes remaining parallel, the new coordinates of the point are (3, 1). Find the old coordinates.
Solution:
New origin (-3, 2) ≡ (h, k)
Let (X, Y) ≡ (3, 1) be the new coordinates and (x, y) be the old coordinates
We have x = X + h and y = Y + k
x = 3 -3 = 0 and Y = 1 + 2 = 3
Ans: The old coordinates of point (3, 1) are (0, 3)
Example – 09:
If the origin is shifted to a point (-3, 2), axes remaining parallel, the new coordinates of the point are (-5, -4). Find the old coordinates.
Solution:
New origin (-3, 2) ≡ (h, k)
Let (X, Y) ≡ (-5, -4) be the new coordinates and (x, y) be the old coordinates
We have x = X + h and y = Y + k
∴ x = -5 -3 = -8 and Y = -4 + 2 = -2
Ans: The old coordinates of point (-5, -4) are (-8, -2)
Example – 10:
If the origin is shifted to a point (-2, 1), axes remaining parallel, the new coordinates of the point are (2, 4). Find the old coordinates.
Solution:
New origin (-2, 1) ≡ (h, k)
Let (X, Y) ≡ (2, 4) be the new coordinates and (x, y) be the old coordinates
We have x = X + h and y = Y + k
∴ x = 2 -2 = 0 and Y = 4 + 1 = 5
Ans: The old coordinates of point (2, 4) are (0, 5)
Example – 11:
If the origin is shifted to a point (-2, 1), axes remaining parallel, the new coordinates of the point are (3, -5). Find the old coordinates.
Solution:
New origin (-2, 1) ≡ (h, k)
Let (X, Y) ≡ (3, -5) be the new coordinates and (x, y) be the old coordinates
We have x = X + h and y = Y + k
∴ x = 3 -2 = 1 and Y = -5 + 1 = -4
Ans: The old coordinates of point (3, -5) are (1, -4)
Example – 12:
If the origin is shifted to a point (-2, 1), axes remaining parallel, the new coordinates of the point are (0, 4). Find the old coordinates.
Solution:
New origin (-2, 1) ≡ (h, k)
Let (X, Y) ≡ (0, 4) be the new coordinates and (x, y) be the old coordinates
We have x = X + h and y = Y + k
x = 0 -2 = -2 and Y = 4 + 1 = 5
Ans: The old coordinates of point (0, 4) are (-2, 5)
Example – 13:
If the origin is shifted to a point (-2, 1), axes remaining parallel, the new coordinates of the point are (-4, 8). Find the old coordinates.
Solution:
New origin (-2, 1) ≡ (h, k)
Let (X, Y) ≡ (-4, 8) be the new coordinates and (x, y) be the old coordinates
We have x = X + h and y = Y + k
∴ x = -4 -2 = -6 and Y = 8 + 1 = 9
Ans: The old coordinates of point (-4, 8) are (-6, 9)
Example – 14:
If the origin is shifted to a point (-2, 1), axes remaining parallel, the new coordinates of the point are (-6, 3). Find the old coordinates.
Solution:
New origin (-2, 1) ≡ (h, k)
Let (X, Y) ≡ (-6, 3) be the new coordinates and (x, y) be the old coordinates
We have x = X + h and y = Y + k
x = -6 -2 = -8 and Y = 3 + 1 = 4
Ans: The old coordinates of point (-6, 3) are (-8, 4)
Type – III: Find Coordinates of Point Where Origin is Shifted (Location of Shift of Origin)
Example – 15:
The point (3, 8) becomes (-2, 1) after the shift of origin. Find the coordinates of the point, where the origin is shifted.
Solution:
Let the new origin be (h, k)
Let (x, y) ≡ (3, 8) be the old coordinates and (X, Y) ≡ (-2, 1) be the new coordinates
We have h = x – X and k = y – Y
∴ h = 3 + 2 = 5 and Y = 8 – 1 = 7
Ans: The coordinates of the point, where the origin is shifted are (5, 7)
Example – 16:
The point (2, 3) becomes (5, -2) after the shift of origin. Find the coordinates of the point, where the origin is shifted.
Solution:
Let the new origin be (h, k)
Let (x, y) ≡ (2, 3) be the old coordinates and (X, Y) ≡ (5, -2) be the new coordinates
We have h = x – X and k = y – Y
∴ h = 2 – 5 = -3 and Y = 3 + 2 = 5
Ans: The coordinates of the point, where the origin is shifted are (-3, 5)
Example – 17:
The point (3, -4) becomes (7, 2) after the shift of origin. Find the coordinates of the point, where the origin is shifted.
Solution:
Let the new origin be (h, k)
Let (x, y) ≡ (3, -4) be the old coordinates and (X, Y) ≡ (7, 2) be the new coordinates
We have h = x – X and k = y – Y
∴ h = 3 – 7 = -4 and Y = -4 -2 = -6
Ans: The coordinates of the point, where the origin is shifted are (-4, -6)
Example – 18:
The point (a, b) becomes (c, d) after the shift of origin. Find the coordinates of the point, where the origin is shifted.
Solution:
Let the new origin be (h, k)
Let (x, y) ≡ (a, b) be the old coordinates and (X, Y) ≡ (c, d) be the new coordinates
We have h = x – X and k = y – Y
∴ h = a – c and Y = b – d
Hence the coordinates of the point, where the origin is shifted are (a – c, b – d)
Miscellaneous:
Example – 19:
The origin is shifted to the point (-5, 9)axes remaining parallel if the point (3, b) lies on the new x-axis and point (a, 3) lies on the new y-axis, find the values of a and b
Solution:
New origin (-5, 9) ≡ (h, k)
Let (x, y) ≡ B(3, b) be the old coordinates and (X, Y) be the new coordinates
We have X = x – h and Y = y – k
∴ X = 3 + 5 = 8 and Y = b – 9
Hence new coordinates of point B are (8, b- 9)
Now point B lies on new x-axis i.e. Y = 0
b – 9 = 0
b = 9
Let (x, y) ≡ A(a, 3) be the old coordinates and (X, Y) be the new coordinates
We have X = x – h and Y = y – k∴ X = a + 5 and Y = 3 – 9 = – 6
Hence new coordinates of point A are (a + 5, – 6)
Now point B lies on new y-axis i.e. X = 0
a + 5 = 0
a = -5
Ans: a = -5 and b = 9