Consider a pair of perpendicular straight lines `ax^(2)+3xy-2y^(2)-5x+5y+c=0`.
The value of c is

Options

A. -3
B. 3
C. -1
D. 1

Explanation

To find the value of \( c \) in the equation of the pair of perpendicular straight lines given by

\[ ax^2 + 3xy - 2y^2 - 5x + 5y + c = 0, \]

we will follow these steps:

### Step 1: Identify coefficients
From the given equation, we can identify the coefficients:
- \( a = a \)
- \( b = 3 \)
- \( c = -2 \)
- \( d = -5 \)
- \( e = 5 \)
- \( f = c \)

### Step 2: Use the condition for perpendicular lines
For the lines to be perpendicular, the condition is:

\[ a + b = 0. \]

Here, \( b = -2 \). Thus, we have:

\[ a + (-2) = 0 \implies a = 2. \]

### Step 3: Set up the determinant condition
The determinant \( \Delta \) for the pair of straight lines must be equal to zero:

\[ \Delta = abc + 2fgh - af^2 - bg^2 - ch^2 = 0. \]

Substituting the known values:
- \( a = 2 \)
- \( b = -2 \)
- \( c = c \)
- \( f = \frac{5}{2} \)
- \( g = -\frac{5}{2} \)
- \( h = \frac{3}{2} \)

### Step 4: Substitute values into the determinant equation
Now, substituting these values into the determinant equation:

\[ \Delta = (2)(-2)(c) + 2\left(\frac{5}{2}\right)\left(-\frac{5}{2}\right)\left(\frac{3}{2}\right) - (2)\left(\frac{5}{2}\right)^2 - (-2)\left(-\frac{5}{2}\right)^2 - c\left(\frac{3}{2}\right)^2 = 0. \]

### Step 5: Simplify the equation
Calculating each term:

1. \( (2)(-2)(c) = -4c \)
2. \( 2\left(\frac{5}{2}\right)\left(-\frac{5}{2}\right)\left(\frac{3}{2}\right) = 2 \cdot \frac{-25}{4} \cdot \frac{3}{2} = -\frac{75}{4} \)
3. \( - (2)\left(\frac{5}{2}\right)^2 = -2 \cdot \frac{25}{4} = -\frac{50}{4} \)
4. \( - (-2)\left(-\frac{5}{2}\right)^2 = -2 \cdot \frac{25}{4} = -\frac{50}{4} \)
5. \( -c\left(\frac{3}{2}\right)^2 = -c \cdot \frac{9}{4} \)

Putting it all together:

\[ -4c - \frac{75}{4} - \frac{50}{4} - \frac{50}{4} - \frac{9c}{4} = 0. \]

### Step 6: Combine like terms
Combine the constants and the terms with \( c \):

\[ -4c - \frac{75 + 50 + 50}{4} - \frac{9c}{4} = 0. \]

This simplifies to:

\[ -4c - \frac{175}{4} - \frac{9c}{4} = 0. \]

### Step 7: Multiply through by 4 to eliminate the fraction
Multiplying the entire equation by 4 gives:

\[ -16c - 175 - 9c = 0. \]

### Step 8: Combine \( c \) terms
Combine the \( c \) terms:

\[ -25c - 175 = 0. \]

### Step 9: Solve for \( c \)
Solving for \( c \):

\[ -25c = 175 \implies c = -\frac{175}{25} = -7. \]

Thus, the value of \( c \) is

\[ \boxed{-7}. \]