Question

The transformed equation of \( x^2 - y^2 + 2x + 4y = 0 \) when the origin is shifted to the point \( (-1,2) \) is:

• A. \( x^2 + y^2 = 1 \)
• B. \( x^2 + 3y^2 = 1 \)
• C. \( x^2 - y^2 + 3 = 0 \)
• D. \( 4x^2 + 9y^2 = 36 \)

Hint:

For shifting origin, replace \( x \) and \( y \) with \( x - h \) and \( y - k \) and simplify the equation.

Answer 1

The Correct Option is C

To find the transformed equation when the origin is shifted to the point \((-1,2)\), we use the transformation of coordinates formula:

\(x = X + h\) and \(y = Y + k\), where \((h, k) = (-1, 2)\).

This means: \(x = X - 1\) and \(y = Y + 2\).

Substitute into the given equation \(x^2 - y^2 + 2x + 4y = 0\):

1. Replace \(x\) with \(X - 1\) and \(y\) with \(Y + 2\).

2. The equation becomes: \((X - 1)^2 - (Y + 2)^2 + 2(X - 1) + 4(Y + 2) = 0\).

3. Expand and simplify:

\((X^2 - 2X + 1) - (Y^2 + 4Y + 4) + 2X - 2 + 4Y + 8 = 0\)

4. Combine like terms:

\(X^2 - Y^2 + 2X - 2X + 4Y - 4Y + 1 - 4 - 2 + 8 = 0\)

5. Simplify further:

\(X^2 - Y^2 + 3 = 0\)

Thus, the transformed equation is:

\(x^2 - y^2 + 3 = 0\)

Answer 2

Step 1: Understanding Coordinate Transformation When the origin is shifted to \( (h, k) \), the transformation is: \[ X = x - h, \quad Y = y - k \] Given the new origin \( (-1,2) \): \[ X = x + 1, \quad Y = y - 2 \]
Step 2: Substituting into the Given Equation The given equation: \[ x^2 - y^2 + 2x + 4y = 0 \] Substituting \( x = X - 1 \) and \( y = Y + 2 \): \[ (X - 1)^2 - (Y + 2)^2 + 2(X - 1) + 4(Y + 2) = 0 \] Expanding: \[ X^2 - 2X + 1 - (Y^2 + 4Y + 4) + 2X - 2 + 4Y + 8 = 0 \]
Step 3: Simplifying the Expression \[ X^2 - Y^2 + 1 - 4 - 2 + 8 = 0 \] \[ X^2 - Y^2 + 3 = 0 \]
Final Answer: \[ X^2 - Y^2 + 3 = 0 \]