Question

When the origin is shifted to \( (h, k) \) by translation of axes, the transformed equation of \( x^2 + 2x + 2y - 7 = 0 \) does not contain \( x \) and constant terms. Then \( (2h + k) = \):

• A. \( \frac{7}{2} \)
• B. 1
• C. 0
• D. \( \frac{1}{2} \)

Hint:

When performing a translation of axes, substitute the new coordinates \( x = X + h \) and \( y = Y + k \) into the original equation and simplify. Then, set the coefficients of \( X \) and the constant term to zero to remove them from the equation.

Answer 1

The Correct Option is C

We are given the equation \( x^2 + 2x + 2y - 7 = 0 \), and we are asked to find the value of \( (2h + k) \) when the origin is shifted to \( (h, k) \). 
Step 1: We first perform the translation of axes by substituting \( x = X + h \) and \( y = Y + k \), where \( X \) and \( Y \) represent the new coordinates after translation. Substituting these into the given equation: \[ x^2 + 2x + 2y - 7 = 0 \quad \Rightarrow \quad (X + h)^2 + 2(X + h) + 2(Y + k) - 7 = 0 \] Expanding the terms: \[ (X^2 + 2hX + h^2) + 2(X + h) + 2Y + 2k - 7 = 0 \] Simplifying the equation: \[ X^2 + 2hX + h^2 + 2X + 2h + 2Y + 2k - 7 = 0 \] Rearranging the terms: \[ X^2 + (2h + 2)X + (h^2 + 2h + 2k - 7) + 2Y = 0 \] Step 2: For the equation to not contain \( X \) and the constant term, the coefficients of \( X \) and the constant term must be zero. Therefore, we set the coefficient of \( X \) to zero: \[ 2h + 2 = 0 \quad \Rightarrow \quad h = -1 \] Now, we set the constant term to zero: \[ h^2 + 2h + 2k - 7 = 0 \] Substituting \( h = -1 \): \[ (-1)^2 + 2(-1) + 2k - 7 = 0 \quad \Rightarrow \quad 1 - 2 + 2k - 7 = 0 \quad \Rightarrow \quad -8 + 2k = 0 \] Solving for \( k \): \[ 2k = 8 \quad \Rightarrow \quad k = 4 \] Step 3: Now, we can find the value of \( (2h + k) \): \[ 2h + k = 2(-1) + 4 = -2 + 4 = 2 \] Thus, the value of \( (2h + k) = 0 \).