# Shift of Origin

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### 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)

#### Example – 19:

• The origin is shifted to the point (-5, 9)axes remaining parallel, if the point (3, b) lies on new x axis and point (a, 3) lies on 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

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