
What is the condition for two lines represented by the equation $a{x^2} + 2hxy + b{y^2} = 0$ to be perpendicular?
A. $ab = - 1$
B. $a = - b$
C. $a = b$
D. $ab = 1$
Answer
161.1k+ views
Hint: A pair of straight lines, passing through the origin, are represented by a general equation of the form $a{x^2} + 2hxy + b{y^2}=0$ . Sum of the slopes of the two lines is given by $\dfrac{{ - 2h}}{b}$ and the product of the slopes is given by $\dfrac{a}{b}$ . The angle between the two lines, $\theta $ , is calculated using the formula $\tan \theta = \left| {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right|$. We will use this formula to derive the condition.
Formula Used: The angle between a pair of straight lines, passing through the origin, represented by $a{x^2} + 2hxy + b{y^2}=0$ is calculated using the formula $\tan \theta = \left| {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right|$ .
Complete step-by-step solution:
Equation of the pair of lines:
$a{x^2} + 2hxy + b{y^2} = 0$
Now, we know that the tangent of the angle between a pair of lines is given by:
$\tan \theta = \left| {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right|$
For a pair of lines to be perpendicular, $\theta = \dfrac{\pi }{2}$ .
Substituting this value, we get:
$\tan \dfrac{\pi }{2} = \left| {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right|$
We know that the value of $\tan \dfrac{\pi }{2}$ is undefined (or infinity), hence,
$a + b = 0$
This gives $a = - b$ .
Hence, for a pair of straight lines, of the form $a{x^2} + 2hxy + b{y^2} = 0$ , to be perpendicular, $a = - b$ .
Thus, the correct option is B.
Note: While calculating the condition required for the given equation of a pair of lines to be perpendicular, just keep in mind to substitute the correct values in the formula. This will avoid any further mistakes.
Formula Used: The angle between a pair of straight lines, passing through the origin, represented by $a{x^2} + 2hxy + b{y^2}=0$ is calculated using the formula $\tan \theta = \left| {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right|$ .
Complete step-by-step solution:
Equation of the pair of lines:
$a{x^2} + 2hxy + b{y^2} = 0$
Now, we know that the tangent of the angle between a pair of lines is given by:
$\tan \theta = \left| {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right|$
For a pair of lines to be perpendicular, $\theta = \dfrac{\pi }{2}$ .
Substituting this value, we get:
$\tan \dfrac{\pi }{2} = \left| {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right|$
We know that the value of $\tan \dfrac{\pi }{2}$ is undefined (or infinity), hence,
$a + b = 0$
This gives $a = - b$ .
Hence, for a pair of straight lines, of the form $a{x^2} + 2hxy + b{y^2} = 0$ , to be perpendicular, $a = - b$ .
Thus, the correct option is B.
Note: While calculating the condition required for the given equation of a pair of lines to be perpendicular, just keep in mind to substitute the correct values in the formula. This will avoid any further mistakes.
Recently Updated Pages
If there are 25 railway stations on a railway line class 11 maths JEE_Main

Minimum area of the circle which touches the parabolas class 11 maths JEE_Main

Which of the following is the empty set A x x is a class 11 maths JEE_Main

The number of ways of selecting two squares on chessboard class 11 maths JEE_Main

Find the points common to the hyperbola 25x2 9y2 2-class-11-maths-JEE_Main

A box contains 6 balls which may be all of different class 11 maths JEE_Main

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

Displacement-Time Graph and Velocity-Time Graph for JEE

Degree of Dissociation and Its Formula With Solved Example for JEE

Free Radical Substitution Mechanism of Alkanes for JEE Main 2025

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced 2025: Dates, Registration, Syllabus, Eligibility Criteria and More

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations

NCERT Solutions for Class 11 Maths In Hindi Chapter 1 Sets

NCERT Solutions for Class 11 Maths Chapter 6 Permutations and Combinations
