Find th point to which the origin has to be shifted to eliminate x and y terms in the equation
` 14x^(2)-4xy+11y^(2)-36x+48y+41=0.`

Answer

To find the point to which the origin has to be shifted to eliminate the x and y terms in the given equation \( 14x^2 - 4xy + 11y^2 - 36x + 48y + 41 = 0 \), we will follow these steps:

### Step 1: Set up the transformation
We will use the transformation:
\[ X = x - h \quad \text{and} \quad Y = y - k \]
where \( (h, k) \) is the new origin we need to find.

### Step 2: Substitute the transformation into the equation
Substituting \( x = X + h \) and \( y = Y + k \) into the equation:
\[ 14(X + h)^2 - 4(X + h)(Y + k) + 11(Y + k)^2 - 36(X + h) + 48(Y + k) + 41 = 0 \]

### Step 3: Expand the equation
Expanding each term:
1. \( 14(X + h)^2 = 14(X^2 + 2hX + h^2) = 14X^2 + 28hX + 14h^2 \)
2. \( -4(X + h)(Y + k) = -4(XY + kX + hY + hk) = -4XY - 4kX - 4hY - 4hk \)
3. \( 11(Y + k)^2 = 11(Y^2 + 2kY + k^2) = 11Y^2 + 22kY + 11k^2 \)
4. \( -36(X + h) = -36X - 36h \)
5. \( 48(Y + k) = 48Y + 48k \)
6. The constant term remains \( +41 \).

Combining all these, we get:
\[ 14X^2 + 11Y^2 - 4XY + (28h - 4k - 36)X + (22k - 4h + 48)Y + (14h^2 - 4hk + 11k^2 - 36h + 48k + 41) = 0 \]

### Step 4: Collect the coefficients of X and Y
To eliminate the x and y terms, we need to set the coefficients of \( X \) and \( Y \) to zero:
1. Coefficient of \( X \): \( 28h - 4k - 36 = 0 \)
2. Coefficient of \( Y \): \( 22k - 4h + 48 = 0 \)

### Step 5: Solve the equations
We now have a system of equations:
1. \( 28h - 4k - 36 = 0 \) (Equation 1)
2. \( 22k - 4h + 48 = 0 \) (Equation 2)

From Equation 1, we can express \( k \) in terms of \( h \):
\[ k = 7h - 9 \]

Substituting \( k \) into Equation 2:
\[ 22(7h - 9) - 4h + 48 = 0 \]
\[ 154h - 198 - 4h + 48 = 0 \]
\[ 150h - 150 = 0 \implies h = 1 \]

Now substituting \( h = 1 \) back into the equation for \( k \):
\[ k = 7(1) - 9 = -2 \]

### Step 6: Conclusion
The new origin to which the origin has to be shifted is \( (h, k) = (1, -2) \).

### Final Answer
The point to which the origin has to be shifted is \( (1, -2) \).

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