Find the point to which the origin should be shifted after a translation of axes so that the following equations will have no first degree terms :
`(i) x^(2)+y^(2)-5x+2y-5=0`
`(ii) x^(2)+y^(2)-4x-8y+3=0`

Answer

To solve the problem of finding the point to which the origin should be shifted after a translation of axes so that the given equations will have no first-degree terms, we will follow these steps:

### Step 1: Analyze the first equation
The first equation is:
\[ x^2 + y^2 - 5x + 2y - 5 = 0 \]
We need to shift the origin to a new point \((h, k)\).

### Step 2: Substitute the new coordinates
After shifting the origin, we substitute \(x = x' + h\) and \(y = y' + k\) into the equation:
\[ (x' + h)^2 + (y' + k)^2 - 5(x' + h) + 2(y' + k) - 5 = 0 \]

### Step 3: Expand the equation
Expanding the equation:
\[ (x'^2 + 2hx' + h^2) + (y'^2 + 2ky' + k^2) - 5x' - 5h + 2y' + 2k - 5 = 0 \]
This simplifies to:
\[ x'^2 + y'^2 + (2h - 5)x' + (2k + 2)y' + (h^2 + k^2 - 5h - 5) = 0 \]

### Step 4: Set coefficients of \(x'\) and \(y'\) to zero
For the equation to have no first-degree terms, the coefficients of \(x'\) and \(y'\) must be zero:
1. \(2h - 5 = 0\)
2. \(2k + 2 = 0\)

### Step 5: Solve for \(h\) and \(k\)
From \(2h - 5 = 0\):
\[ 2h = 5 \implies h = \frac{5}{2} \]
From \(2k + 2 = 0\):
\[ 2k = -2 \implies k = -1 \]

### Step 6: Conclusion for the first equation
Thus, the origin should be shifted to the point:
\[ \left(\frac{5}{2}, -1\right) \]

### Step 7: Analyze the second equation
The second equation is:
\[ x^2 + y^2 - 4x - 8y + 3 = 0 \]
We will repeat the same process.

### Step 8: Substitute the new coordinates
Substituting \(x = x' + h\) and \(y = y' + k\):
\[ (x' + h)^2 + (y' + k)^2 - 4(x' + h) - 8(y' + k) + 3 = 0 \]

### Step 9: Expand the equation
Expanding gives:
\[ (x'^2 + 2hx' + h^2) + (y'^2 + 2ky' + k^2) - 4x' - 4h - 8y' - 8k + 3 = 0 \]
This simplifies to:
\[ x'^2 + y'^2 + (2h - 4)x' + (2k - 8)y' + (h^2 + k^2 - 4h - 8k + 3) = 0 \]

### Step 10: Set coefficients of \(x'\) and \(y'\) to zero
Setting the coefficients to zero:
1. \(2h - 4 = 0\)
2. \(2k - 8 = 0\)

### Step 11: Solve for \(h\) and \(k\)
From \(2h - 4 = 0\):
\[ 2h = 4 \implies h = 2 \]
From \(2k - 8 = 0\):
\[ 2k = 8 \implies k = 4 \]

### Step 12: Conclusion for the second equation
Thus, the origin should be shifted to the point:
\[ (2, 4) \]

### Final Answer
The points to which the origin should be shifted are:
1. For the first equation: \(\left(\frac{5}{2}, -1\right)\)
2. For the second equation: \((2, 4)\)