Question

The equation $12{x^2} + 7xy + a{y^2} + 13x - y + 3 = 0$ represents a pair of perpendicular lines. Then the value of ‘a’ is ____
A. $\dfrac{7}{2}$
B. $ - 19$
C. $ - 12$
D. $12$

Answer 1

Hint: In order to solve this type of question, first we need to write the given equation and the general equation. Then, we have to compare both the equations to get the coefficient of ${x^2}$ and ${y^2}$. Next, we will use the property of perpendicular lines and substitute the values of coefficients obtained in it to get the correct answer.

Formula used:
$a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$
For perpendicular lines,
Coefficient of ${x^2} + $ coefficient of ${y^2} = 0$

Complete step by step solution:
We are given that,
$12{x^2} + 7xy + a{y^2} + 13x - y + 3 = 0$ ………………….equation$\left( 1 \right)$
We know that the second order general equation is,
$a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$ ………………….equation$\left( 2 \right)$
On comparing equation $\left( 1 \right)$ and $\left( 2 \right)$ we get,
$a = 12,\;b = a$
We also know that,
For perpendicular lines,
Coefficient of ${x^2} + $ coefficient of ${y^2} = 0$
$ \Rightarrow a + b = 0$
Substituting the values of a and b,
$ \Rightarrow 12 + a = 0$
$ \Rightarrow a = - 12$
$\therefore $ The correct option is C.

Note: To solve this type of question use the property mentioned in the question ( i.e., here, property of perpendicular lines) in order to get the desired correct answer. Also, one has to be very sure while writing the general equation and comparing the coefficients.