Question

If the equation $ax^2 + 3xy - 2y^2 - 5x + 5y + c = 0$ represents a pair of perpendicular lines, the value of a + c is

• A. 1
• B. -1
• C. 2
• D. -2

Answer 1

Answer: (B)

-1

Answer 2

The general equation of a pair of straight lines is $Ax^2 + 2Hxy + By^2 + 2Gx + 2Fy + C = 0$.

Given equation: $ax^2 + 3xy - 2y^2 - 5x + 5y + c = 0$.

Comparing coefficients: $A = a, H = 3/2, B = -2, G = -5/2, F = 5/2, C = c$.

For the lines to be perpendicular, $A + B = 0$. $a + (-2) = 0 \implies a = 2$.

For the equation to represent a pair of straight lines, the determinant $\Delta = ABC + 2FGH - AF^2 - BG^2 - CH^2 = 0$.

Substitute the values: $(2)(-2)(c) + 2(5/2)(-5/2)(3/2) - (2)(5/2)^2 - (-2)(-5/2)^2 - (c)(3/2)^2 = 0$ $-4c + 2(-75/8) - 2(25/4) + 2(25/4) - c(9/4) = 0$ $-4c - 75/4 - 25/2 + 25/2 - 9c/4 = 0$ $-4c - 75/4 - 9c/4 = 0$

Multiply by 4: $-16c - 75 - 9c = 0$ $-25c = 75$ $c = -3$.

We need to find $a + c = 2 + (-3) = -1$.