Show that the equation `a(x^4+y^4)-4bxy(x^2-y^2)+6cx^2y^2=0` represents two pairs of lines at right angles and that `if 2b^2=a^2+3ac` , the two pairs will coincide.

Answer

To show that the equation \( a(x^4 + y^4) - 4bxy(x^2 - y^2) + 6cx^2y^2 = 0 \) represents two pairs of lines at right angles, we can follow these steps:

### Step 1: Rewrite the given equation
We start with the given equation:
\[ a(x^4 + y^4) - 4bxy(x^2 - y^2) + 6cx^2y^2 = 0 \]

### Step 2: Factor the equation
To show that this equation represents two pairs of lines, we can factor it. We can express \( x^4 + y^4 \) as \( (x^2 + y^2)^2 - 2x^2y^2 \):
\[ x^4 + y^4 = (x^2 + y^2)^2 - 2x^2y^2 \]
Substituting this into the equation gives:
\[ a((x^2 + y^2)^2 - 2x^2y^2) - 4bxy(x^2 - y^2) + 6cx^2y^2 = 0 \]
This simplifies to:
\[ a(x^2 + y^2)^2 + (6c - 2a)x^2y^2 - 4bxy(x^2 - y^2) = 0 \]

### Step 3: Identify conditions for pairs of lines
For the equation to represent two pairs of lines at right angles, the sum of the coefficients of \( x^2 \) and \( y^2 \) must equal zero. The coefficients can be identified from the factored form:
- Coefficient of \( x^2 \): \( a + 6c - 2a = 6c - a \)
- Coefficient of \( y^2 \): \( a + 6c - 2a = 6c - a \)

Setting the sum equal to zero:
\[ (6c - a) + (6c - a) = 0 \implies 12c - 2a = 0 \implies 6c = a \]

### Step 4: Check for right angles
Next, we need to ensure that the lines are at right angles. This can be checked using the condition that the determinant of the coefficients of \( x^2, xy, y^2 \) must be zero:
\[ \begin{vmatrix} a & -2b & -a \\ -a & 0 & 2b \\ 0 & -2b & a \end{vmatrix} = 0 \]

Calculating this determinant will yield a condition that must hold true for the lines to be perpendicular.

### Step 5: Coincidence condition
Now, we need to show that if \( 2b^2 = a^2 + 3ac \), the two pairs of lines will coincide. For coinciding lines, the discriminant of the quadratic formed by the coefficients must be zero. This leads to:
\[ (2b)^2 = 4ab \implies 4b^2 = 4ac \implies b^2 = ac \]
Thus, we can conclude that the two pairs of lines coincide under the given condition.

### Final Result
Thus, we have shown that the equation represents two pairs of lines at right angles, and if \( 2b^2 = a^2 + 3ac \), the two pairs will coincide.