Shifting of Origin MCQ Quiz - Objective Question with Answer for Shifting of Origin - Download Free PDF

Calculation

Given:

Original point: (-1, 2)

New origin: (2, -1)

Rotation angle: 45°

Reflection through y = x

1) Translation of axes:

Let the new coordinates be (a, b).

\(a = x - h = -1 - 2 = -3\)

\(b = y - k = 2 - (-1) = 3\)

⇒ \((a, b) = (-3, 3)\)

2) Rotation of axes by 45°:

Let the new coordinates be (c, d).

\(c = a\cos\theta + b\sin\theta = -3\cos 45° + 3\sin 45° = -3(\frac{1}{\sqrt{2}}) + 3(\frac{1}{\sqrt{2}}) = 0\)

\(d = -a\sin\theta + b\cos\theta = -(-3)\sin 45° + 3\cos 45° = 3(\frac{1}{\sqrt{2}}) + 3(\frac{1}{\sqrt{2}}) = \frac{6}{\sqrt{2}} = 3\sqrt{2}\)

⇒ \((c, d) = (0, 3\sqrt{2})\)

3) Reflection through y = x:

Let the new coordinates be (e, f).

When reflected through y = x, (x, y) becomes (y, x).

⇒ \((e, f) = (3\sqrt{2}, 0)\)

Hence option 3 is correct

Concept:

Shifting the Origin by Translation of Axes:

• In this problem, we are required to shift the origin to a new point P using the translation of axes in order to remove the linear y-term from the given equation.
• The general form of the equation is x² - y² + 2y - 1 = 0, and we are tasked with finding the transformed equation after shifting the origin.

Calculation:

Given the equation:

x² - y² + 2y - 1 = 0

Step 1: Completing the Square

To eliminate the linear y-term, we complete the square:

x² - (y² - 2y) - 1 = 0

Rewriting y² - 2y:

y² - 2y = (y - 1)² - 1

Thus, substituting back:

x² - ((y - 1)² - 1) - 1 = 0

Step 2: Shifting the Origin

Introduce a new coordinate system Y = y - 1, where the origin is shifted to (0, 1). The transformed equation becomes:

x² - Y² = 0

Since Y = y - 1, we conclude:

x² - y² = 0

Thus, the transformed equation is:

x² - y² = 0

Concept Used:

Translation of Axes: If the origin is shifted to (h, k), the new coordinates (X, Y) of a point (x, y) are given by X = x - h and Y = y - k.

General Equation of second degree: \(AX^2 + 2HXY + BY^2 + 2GX + 2FY + C = 0\)

Calculation:

Given:

When the origin is shifted to (a, b), the point (4, 4) becomes (6, 8).

The transformed equation of \(x^2 + 4xy + y^2 = 0\) is \(X^2 + 2HXY + Y^2 + 2GX + 2FY + C = 0\)

From the provided solution:

x = X + a = X + 8

y = Y + b = Y - 4

Transformed equation: \((x+8)^2 + 4(x+8)(y-4) + (y-4)^2 = 0\)

Comparing coefficients: H = 2, G = 0, F = 12, C = -48

⇒ 2H(G + F) = 2 × 2 × (0 + 12) = 48

⇒ Since C = -48,

∴ 2H(G + F) = -C

Hence option 4 is correct.

We will update the solution 

Concept: 

If (x , y) are the coordinates of a point and the origin is shifted to (h, k), then the new coordinates (X, Y) of the point will be given by

x = X + h and y = Y + k

Explanation: 

Given, x2 + y2 + 2x - 4y + 1 = 0

⇒ x² + 2x + 1 + y² - 4y + 4 - 4 = 0

⇒ (x + 1)² + ( y - 2)² = 4

The origin is shifted to (-1, 2) by the translation of axes,

∴  x = X - 1 and y = Y + 2 

The transformed equation will be

(X - 1 + 1)² + ( Y + 2 - 2)² = 4

∴ X² + Y² = 4

The correct answer is option (1).

Concept: 

If (x , y) are the coordinates of a point and the origin is shifted to (h, k), then the new coordinates (X, Y) of the point will be given by

x = X + h and y = Y + k

Explanation: 

Given, x2 + y2 + 2x - 4y + 1 = 0

⇒ x² + 2x + 1 + y² - 4y + 4 - 4 = 0

⇒ (x + 1)² + ( y - 2)² = 4

The origin is shifted to (-1, 2) by the translation of axes,

∴  x = X - 1 and y = Y + 2 

The transformed equation will be

(X - 1 + 1)² + ( Y + 2 - 2)² = 4

∴ X² + Y² = 4

The correct answer is option (1).

Calculation

Given:

Original point: (-1, 2)

New origin: (2, -1)

Rotation angle: 45°

Reflection through y = x

1) Translation of axes:

Let the new coordinates be (a, b).

\(a = x - h = -1 - 2 = -3\)

\(b = y - k = 2 - (-1) = 3\)

⇒ \((a, b) = (-3, 3)\)

2) Rotation of axes by 45°:

Let the new coordinates be (c, d).

\(c = a\cos\theta + b\sin\theta = -3\cos 45° + 3\sin 45° = -3(\frac{1}{\sqrt{2}}) + 3(\frac{1}{\sqrt{2}}) = 0\)

\(d = -a\sin\theta + b\cos\theta = -(-3)\sin 45° + 3\cos 45° = 3(\frac{1}{\sqrt{2}}) + 3(\frac{1}{\sqrt{2}}) = \frac{6}{\sqrt{2}} = 3\sqrt{2}\)

⇒ \((c, d) = (0, 3\sqrt{2})\)

3) Reflection through y = x:

Let the new coordinates be (e, f).

When reflected through y = x, (x, y) becomes (y, x).

⇒ \((e, f) = (3\sqrt{2}, 0)\)

Hence option 3 is correct

Concept:

Shifting the Origin by Translation of Axes:

• In this problem, we are required to shift the origin to a new point P using the translation of axes in order to remove the linear y-term from the given equation.
• The general form of the equation is x² - y² + 2y - 1 = 0, and we are tasked with finding the transformed equation after shifting the origin.

Calculation:

Given the equation:

x² - y² + 2y - 1 = 0

Step 1: Completing the Square

To eliminate the linear y-term, we complete the square:

x² - (y² - 2y) - 1 = 0

Rewriting y² - 2y:

y² - 2y = (y - 1)² - 1

Thus, substituting back:

x² - ((y - 1)² - 1) - 1 = 0

Step 2: Shifting the Origin

Introduce a new coordinate system Y = y - 1, where the origin is shifted to (0, 1). The transformed equation becomes:

x² - Y² = 0

Since Y = y - 1, we conclude:

x² - y² = 0

Thus, the transformed equation is:

x² - y² = 0

Concept Used:

Translation of Axes: If the origin is shifted to (h, k), the new coordinates (X, Y) of a point (x, y) are given by X = x - h and Y = y - k.

General Equation of second degree: \(AX^2 + 2HXY + BY^2 + 2GX + 2FY + C = 0\)

Calculation:

Given:

When the origin is shifted to (a, b), the point (4, 4) becomes (6, 8).

The transformed equation of \(x^2 + 4xy + y^2 = 0\) is \(X^2 + 2HXY + Y^2 + 2GX + 2FY + C = 0\)

From the provided solution:

x = X + a = X + 8

y = Y + b = Y - 4

Transformed equation: \((x+8)^2 + 4(x+8)(y-4) + (y-4)^2 = 0\)

Comparing coefficients: H = 2, G = 0, F = 12, C = -48

⇒ 2H(G + F) = 2 × 2 × (0 + 12) = 48

⇒ Since C = -48,

∴ 2H(G + F) = -C

Hence option 4 is correct.

We will update the solution 

 Concept:

Shifting of origin:

Let (x, y)  be the coordinates of a point, and the origin is shifted to point (h, k) without changing the orientation of the coordinate axes, then the new  coordinates (X, Y) of the point will be given by X = x - h and Y = y - k.

Thus we have x = X + h and y =  Y + k.

Explanation:

The given equation is 

Since the origin is shifted to the point \(\left(\frac{3}{2},\frac{3}{2}\right)\)  by the translation of coordinate axes.

Therefore by replacing x by x = X + \(\frac{3}{2}\) and  y = Y + \(\frac{3}{2}\), we get

32( X + \(\frac{3}{2}\) )² + 8( X + \(\frac{3}{2}\) )(Y + \(\frac{3}{2}\)) + 32(Y + \(\frac{3}{2}\))² − 108( X + \(\frac{3}{2}\) ) − 108(Y + \(\frac{3}{2}\)) + 99 = 0

On solving, we get

32X² + 8XY + 32Y² - 63 = 0

which is the required equation.

Hence option (4) is correct.

Concept:

• Given axes are rotated by angle θ through counter-clockwise direction:

  ⇒ X' = x cos θ - y sin θ  and  Y' = x sin θ  + y cos θ .

Calculation:

Given axes are rotated by angle θ through counter-clockwise direction:

⇒ X' = x cos θ - y sin θ  and  Y' = x sin θ  + y cos θ .

Given the origin is shifted to the point (2, 3).

⇒ X = 2 +  x cos θ - y sin θ  and  Y = 3 + x sin θ  + y cos θ.

∴ Given equation 3x2 + 2xy + 3y2 - 18x - 22y + 50 = 0 in new co-ordinate can be written as:

3(2 +  x cos θ - y sin θ)² + 2(2 +  x cos θ - y sin θ)(3 + x sin θ  + y cos θ)+ 3(3 + x sin θ  + y cos θ)² 

- 18(2 +  x cos θ - y sin θ) - 22(3 + x sin θ  + y cos θ) +50 = 0       ...(1)

Given after transformation, the new equation is 4x2 + 2y2 - 1 = 0

∵ xy term in the new equation is missing, put xy term of equation (1) to zero.

⇒ - 3xy sin 2θ + xy cos² θ - xy sin² θ + 3xy sin 2θ   = 0 

⇒ xy cos 2θ = 0

⇒ 2θ = \(\frac{\pi}{2}\)

⇒ θ = \(\frac{\pi}{4}\).

The required value of θ = \(\frac{\pi}{4}\)