The y-axis and the lines `(a^(5)-2a^(3))x+(a+2)y+3a=0` and `(a^(5)-3a^(2))x+4y+a-2=0` are concurrent for

Options

A. Two values of a
B. Three values of a
C. Five values of a
D. no value of a

Answer

To determine the values of \( a \) for which the y-axis and the lines given by the equations

1. \( (a^5 - 2a^3)x + (a + 2)y + 3a = 0 \)
2. \( (a^5 - 3a^2)x + 4y + (a - 2) = 0 \)

are concurrent, we need to find the intersection point of these lines with the y-axis (where \( x = 0 \)) and set the y-coordinates equal to each other.

### Step 1: Substitute \( x = 0 \) into the first line's equation.

Substituting \( x = 0 \) into the first line:

\[ (a^5 - 2a^3)(0) + (a + 2)y + 3a = 0 \]

This simplifies to:

\[ (a + 2)y + 3a = 0 \]

Rearranging gives:

\[ (a + 2)y = -3a \]

Thus,

\[ y = \frac{-3a}{a + 2} \quad \text{(1)} \]

### Step 2: Substitute \( x = 0 \) into the second line's equation.

Now substituting \( x = 0 \) into the second line:

\[ (a^5 - 3a^2)(0) + 4y + (a - 2) = 0 \]

This simplifies to:

\[ 4y + (a - 2) = 0 \]

Rearranging gives:

\[ 4y = - (a - 2) \]

Thus,

\[ y = \frac{2 - a}{4} \quad \text{(2)} \]

### Step 3: Set the two expressions for \( y \) equal to each other.

From equations (1) and (2), we have:

\[ \frac{-3a}{a + 2} = \frac{2 - a}{4} \]

### Step 4: Cross-multiply to eliminate the fractions.

Cross-multiplying gives:

\[ -3a \cdot 4 = (2 - a)(a + 2) \]

This simplifies to:

\[ -12a = 2a + 4 - a^2 - 2a \]

Combining like terms results in:

\[ -12a = 4 - a^2 \]

### Step 5: Rearrange the equation.

Rearranging gives:

\[ a^2 - 12a + 4 = 0 \]

### Step 6: Solve the quadratic equation using the quadratic formula.

Using the quadratic formula \( a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

Here, \( a = 1, b = -12, c = 4 \):

\[ a = \frac{12 \pm \sqrt{(-12)^2 - 4 \cdot 1 \cdot 4}}{2 \cdot 1} \]

Calculating the discriminant:

\[ = \frac{12 \pm \sqrt{144 - 16}}{2} \]
\[ = \frac{12 \pm \sqrt{128}}{2} \]
\[ = \frac{12 \pm 8\sqrt{2}}{2} \]
\[ = 6 \pm 4\sqrt{2} \]

Thus, the values of \( a \) for which the lines are concurrent are:

\[ a = 6 + 4\sqrt{2} \quad \text{and} \quad a = 6 - 4\sqrt{2} \]