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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: y2 + x2 − 4x − 8y + 3 = 0
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Solution
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 y2 + x2 − 4x − 8y + 3 = 0, we get:
\[\left( Y + k \right)^2 + \left( X + h \right)^2 - 4\left( X + h \right) - 8\left( Y + k \right) + 3 = 0\]
\[ \Rightarrow Y^2 + 2kY + k^2 + X^2 + 2hX + h^2 - 4X - 4h - 8Y - 8k + 3 = 0\]
\[ \Rightarrow X^2 + Y^2 + X\left( 2h - 4 \right) + Y\left( 2k - 8 \right) + k^2 + h^2 - 4h - 8k + 3 = 0\]
For this equation to be free from the terms containing X and Y, we must have \[2h - 4 = 0, 2k - 8 = 0\]
\[ \Rightarrow h = 2, k = 4\]
Hence, the origin should be shifted to the point (2, 4).
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