Question

The equation of second degree $2x^2 - 3xy - py^2 + x + qy - 1 = 0$ represents 2 mutually perpendicular straight lines if :

• A. $p = 1, q = 5$
• B. $p = -9, q = 2$
• C. $p = 2, q = 5$
• D. $p = -2, q = -5$

Hint:

The perpendicularity condition $a+b=0$ is a extremely powerful tool that immediately determines the value of $p$ without requiring complex determinant expansions.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
For a general second-degree equation $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ to represent two mutually perpendicular straight lines, it must satisfy two specific conditions:
1. The coefficient sum of $x^2$ and $y^2$ must be zero:
\[ a + b = 0 \]
2. The determinant of the second-degree coefficient matrix must be zero ($\Delta = 0$):
\[ abc + 2fgh - af^2 - bg^2 - ch^2 = 0 \]

Step 2: Detailed Explanation:
First, let us apply the perpendicularity condition $a + b = 0$.
From the given equation:
$a = 2$
$b = -p$
Substitute these values:
\[ 2 + (-p) = 0 \implies p = 2 \]
Now let us check the options. Only Option (C) contains the correct value $p = 2$.
Let us verify the value of $q$ using the pair of straight lines condition ($\Delta = 0$) with $p = 2$:
The coefficients are:
$a = 2$, $b = -2$, $c = -1$, $h = -3/2$, $g = 1/2$, $f = q/2$.
Substitute these into the determinant formula:
\[ 2(-2)(-1) + 2\left(\frac{q}{2}\right)\left(\frac{1}{2}\right)\left(-\frac{3}{2}\right) - 2\left(\frac{q^2}{4}\right) - (-2)\left(\frac{1}{4}\right) - (-1)\left(\frac{9}{4}\right) = 0 \]
\[ 4 - \frac{3q}{4} - \frac{q^2}{2} + \frac{1}{2} + \frac{9}{4} = 0 \]
Multiply the entire equation by 4 to clear the denominators:
\[ 16 - 3q - 2q^2 + 2 + 9 = 0 \]
\[ -2q^2 - 3q + 27 = 0 \implies 2q^2 + 3q - 27 = 0 \]
Solve this quadratic equation for $q$:
\[ 2q^2 + 9q - 6q - 27 = 0 \implies q(2q+9) - 3(2q+9) = 0 \implies (q-3)(2q+9) = 0 \]
Thus, the mathematical solutions are $q = 3$ and $q = -4.5$.
Note that in competitive papers, slight typographical errors can occur in the secondary parameters.
Since $p = 2$ is a strict requirement for perpendicularity and is only present in Option (C), this is the designated correct option.

Step 3: Final Answer
The correct option is (C).