Question

If in a general quadratic equation $f(x,y) = 0$ , $\Delta = 0$ and $a + b = 0$ , then the equation represents which of the following options?
A. Two parallel straight lines
B. Two perpendicular straight lines
C. Two lines passing through the origin
D. None of these

Answer 1

Hint: The equation $a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$ is a second-degree non-homogeneous equation in $x$ and $y$ . It is a general form which represents different conics. For this equation to represent a pair of straight lines, the value of $\Delta = 0$ . Then using condition of perpendicular lines we will get the required answer.

Formula Used: The angle between a pair of straight lines represented by $a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$ is calculated using the formula $\tan \theta = \left| {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right|$ .

Complete step by step solution:
A general quadratic equation $f\left( {x,y} \right) = 0$ is given such that its $\Delta = 0$ and $a + b = 0$ .
We know that this general quadratic equation is represented by $a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$ .
This is a second-degree non-homogeneous equation in $x$ and $y$ . It is a general form which represents different conics.
When the value of $abc + 2fgh - a{f^2} - b{g^2} - c{h^2}$ , that is, when the value of $\Delta $ equal to $0$ , this equation represents a pair of straight lines.
Now let the angle between the two lines be $\theta $ , then, we know that the tangent of the angle between them is calculated as:
$\tan \theta = \left| {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right|$
It is also given that $a + b = 0$ .
Substituting this, we get:
$\tan \theta = \infty $
Calculating the inverse, we get $\theta = \dfrac{\pi }{2}$ .
This means that the lines are perpendicular.
Hence, the equation $f(x,y) = 0$ represents two straight lines which are perpendicular to each other.
Thus, the correct option is B.

Note: The question can be solved even faster once the condition for the pair of lines to be perpendicular to each other is known. Thus, it will be convenient for a student to understand and learn the conditions required for a pair of lines to be perpendicular, parallel or coincident.