Find th transformed equation of
`2x^(2) +4xy +5y^(2) =0` when the origin is shifted to the point (3,4).

Answer

To find the transformed equation of \(2x^2 + 4xy + 5y^2 = 0\) when the origin is shifted to the point (3, 4), we will follow these steps:

### Step 1: Define the new coordinates
We define the new coordinates \(X\) and \(Y\) in terms of the old coordinates \(x\) and \(y\):
\[ X = x - 3 \]
\[ Y = y - 4 \]

### Step 2: Substitute the new coordinates into the original equation
We will substitute \(x\) and \(y\) in terms of \(X\) and \(Y\):
\[ x = X + 3 \]
\[ y = Y + 4 \]
Now, substitute these into the original equation:
\[ 2(X + 3)^2 + 4(X + 3)(Y + 4) + 5(Y + 4)^2 = 0 \]

### Step 3: Expand the equation
Now we will expand each term:
1. Expand \(2(X + 3)^2\):
\[ 2(X^2 + 6X + 9) = 2X^2 + 12X + 18 \]

2. Expand \(4(X + 3)(Y + 4)\):
\[ 4(XY + 4X + 3Y + 12) = 4XY + 16X + 12Y + 48 \]

3. Expand \(5(Y + 4)^2\):
\[ 5(Y^2 + 8Y + 16) = 5Y^2 + 40Y + 80 \]

### Step 4: Combine all the expanded terms
Now we combine all the expanded terms:
\[ 2X^2 + 12X + 18 + 4XY + 16X + 12Y + 48 + 5Y^2 + 40Y + 80 = 0 \]

### Step 5: Group similar terms
Now, we will group similar terms together:
- \(2X^2\)
- \(4XY\)
- \(5Y^2\)
- Combine \(X\) terms: \(12X + 16X = 28X\)
- Combine \(Y\) terms: \(12Y + 40Y = 52Y\)
- Combine constant terms: \(18 + 48 + 80 = 146\)

Putting it all together, we get:
\[ 2X^2 + 4XY + 5Y^2 + 28X + 52Y + 146 = 0 \]

### Step 6: Write the final transformed equation
Finally, we replace \(X\) and \(Y\) back with \(x\) and \(y\):
\[ 2x^2 + 4xy + 5y^2 + 28x + 52y + 146 = 0 \]

### Final Answer:
The transformed equation is:
\[ 2x^2 + 4xy + 5y^2 + 28x + 52y + 146 = 0 \]