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

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: $$X^2 - Y^2 + 3 = 0$$