The equation of the straigh lines through the point `(x_(1),y_(1))` and parallel to the lines given by `ax^(2)+2xy+ay^(2)=0`, is

Options

A. `a(y-y_(1))^(2)+2h(x-x_(1))(y-y_(1))+b(x-x_(1))^(2)=0`
B. `a(y-y_(1))^(2)-2h(x-x_(1))(y-y_(1))+b(x-x_(1))^(2)=0`
C. `b(y-y_(1))^(2)+2h(x-x_(1))(y-y_(1))+a(x-x_(1))^(2)=0`
D. none of these

Explanation

To find the equation of the straight lines through the point \((x_1, y_1)\) and parallel to the lines given by the equation \(ax^2 + 2xy + ay^2 = 0\), we can follow these steps:

### Step 1: Understand the given equation
The equation \(ax^2 + 2xy + ay^2 = 0\) represents a pair of straight lines through the origin. We need to find lines that are parallel to these lines and pass through the point \((x_1, y_1)\).

### Step 2: Rewrite the equation in terms of shifted coordinates
To find the equation of the lines parallel to the given lines and passing through \((x_1, y_1)\), we can shift the origin to this point. We can substitute \(x = X + x_1\) and \(y = Y + y_1\) into the original equation, where \(X\) and \(Y\) are the new coordinates.

### Step 3: Substitute into the original equation
Substituting \(x = X + x_1\) and \(y = Y + y_1\) into the equation \(ax^2 + 2xy + ay^2 = 0\), we get:
\[ a(X + x_1)^2 + 2(X + x_1)(Y + y_1) + a(Y + y_1)^2 = 0 \]

### Step 4: Expand the equation
Expanding this equation:
\[ a(X^2 + 2Xx_1 + x_1^2) + 2(XY + Xy_1 + Yx_1 + x_1y_1) + a(Y^2 + 2Yy_1 + y_1^2) = 0 \]
This simplifies to:
\[ aX^2 + 2XY + aY^2 + 2Xx_1 + 2Yy_1 + ax_1^2 + 2x_1y_1 + ay_1^2 = 0 \]

### Step 5: Rearranging the equation
Rearranging the equation gives us:
\[ aX^2 + 2XY + aY^2 + 2Xx_1 + 2Yy_1 + (ax_1^2 + 2x_1y_1 + ay_1^2) = 0 \]
Since we want the lines through \((x_1, y_1)\), we can set the constant term to zero:
\[ aX^2 + 2XY + aY^2 + 2Xx_1 + 2Yy_1 = 0 \]

### Step 6: Final form of the equation
The final equation of the straight lines through the point \((x_1, y_1)\) and parallel to the lines given by \(ax^2 + 2xy + ay^2 = 0\) is:
\[ a(x - x_1)^2 + 2(x - x_1)(y - y_1) + a(y - y_1)^2 = 0 \]

### Conclusion
Thus, the required equation is:
\[ a(x - x_1)^2 + 2(x - x_1)(y - y_1) + a(y - y_1)^2 = 0 \]