Equation `ax^3-9x^2y-xy^2+4y^3=0` represents three straight lines. If the two of the lines are perpendicular , then a is equal to

Options

A. -5
B. 5
C. -4
D. 4

Answer

To find the value of \( a \) such that the equation \( ax^3 - 9x^2y - xy^2 + 4y^3 = 0 \) represents three straight lines, two of which are perpendicular, we can follow these steps:

### Step 1: Identify the given equation
The equation is given as:
\[ ax^3 - 9x^2y - xy^2 + 4y^3 = 0 \]

### Step 2: Rewrite the equation
We can express the equation in a more manageable form, focusing on the terms that involve \( x \) and \( y \):
\[ -xy^2 + 4y^3 = 0 \]

### Step 3: Use the condition for perpendicular lines
For the two lines represented by the equation to be perpendicular, the sum of the coefficients of \( x \) and \( y \) must equal zero.

### Step 4: Assume a general form for the lines
We can assume a general form for the equation of two lines:
\[ x^2 + 2hxy - y^2 = 0 \]
Here, the coefficients of \( x \) and \( y^2 \) are \( 1 \) and \( -1 \) respectively.

### Step 5: Divide the original equation by the assumed form
Dividing the original equation by the assumed form gives us:
\[ \frac{ax^3 - 9x^2y - xy^2 + 4y^3}{x^2 + 2hxy - y^2} \]

### Step 6: Collect terms and compare coefficients
After performing polynomial long division, we can express the result in terms of \( x \) and \( y \):
\[ ax^3 - (4 + h)x^2y + (4 + a)y^3 \]

### Step 7: Compare coefficients
From the original equation, we compare coefficients:
1. Coefficient of \( x^2y \):
\[ -9 = 4 + h \implies h = -13 \]
2. Coefficient of \( xy^2 \):
\[ -1 = -4h - a \]

### Step 8: Substitute \( h \) into the second equation
Substituting \( h = -13 \) into the second equation:
\[ -1 = -4(-13) - a \implies -1 = 52 - a \implies a = 53 \]

### Step 9: Conclusion
Thus, the value of \( a \) for which the two lines are perpendicular is:
\[ \boxed{53} \]

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