Question

Find the point to which the origin should be shifted after a translation of axes so that the following equation will have no first degree terms: x² + y² − 5x + 2y − 5 = 0

Answer 1

Let the origin be shifted to (h, k). Then, x = X + h and y = Y + k.
Substituting x = X + h and y = Y + k in the equation x² + y² − 5x + 2y − 5 = 0, we get:
\[\left( X + h \right)^2 + \left( Y + k \right)^2 - 5\left( X + h \right) + 2\left( Y + k \right) - 5 = 0\]
\[ \Rightarrow X^2 + 2hX + h^2 + Y^2 + 2kY + k^2 - 5X - 5h + 2Y + 2k - 5 = 0\]
\[ \Rightarrow X^2 + Y^2 + X\left( 2h - 5 \right) + Y\left( 2k + 2 \right) + k^2 + h^2 - 5h + 2k - 5 = 0\]
For this equation to be free from the terms containing X and Y, we must have
\[2h - 5 = 0, 2k + 2 = 0\]
\[ \Rightarrow h = \frac{5}{2}, k = - 1\]
Hence, the origin should be shifted to the point \[\left( \frac{5}{2}, - 1 \right)\]