
The angle between the lines ${x^2} + 4xy + {y^2} = 0$ is
A. ${15^ \circ }$
B. ${60^ \circ }$
C. ${45^ \circ }$
D. ${30^ \circ }$
Answer
161.4k+ views
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 values of a, h and b. Next, we will use the formula for finding the angle between the lines and substitute the values in it to get the correct answer.
Formula used:
$a{x^2} + 2hxy + b{y^2} = 0$
$\tan \theta = \left| {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right|$
Complete step by step solution:
We are given the following pair of lines,
${x^2} + 4xy + {y^2} = 0$ ………………….equation$\left( 1 \right)$
We know that the general equation is,
$a{x^2} + 2hxy + b{y^2} = 0$ ………………….equation$\left( 2 \right)$
On comparing equation $\left( 1 \right)$ and $\left( 2 \right)$ we get,
$a = 1,\;h = 2,\;b = 1$
Finding the angle between the lines,
$\tan \theta = \left| {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right|$
Substituting the values of a, h and b,
$\tan \theta = \left| {\dfrac{{2\sqrt {{{\left( {2} \right)}^2} - 1 \times 1} }}{{1 + 1}}} \right|$
$\tan \theta = \left| {\dfrac{{2\sqrt {4 - 1} }}{2}} \right|$
Solving it,
$\tan \theta = \left| {\dfrac{{2\sqrt 3 }}{2}} \right|$
$\tan \theta = \sqrt 3 $
Solving for $\theta ,$ we get,
$\theta = {60^ \circ }$
$\therefore $ The correct option is B.
Note: To solve this type of question one needs to be careful while writing the general equation and comparing the values of a, h and b. Avoid any type of calculation mistakes while substituting and solving the equation to get the correct angle.
Formula used:
$a{x^2} + 2hxy + b{y^2} = 0$
$\tan \theta = \left| {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right|$
Complete step by step solution:
We are given the following pair of lines,
${x^2} + 4xy + {y^2} = 0$ ………………….equation$\left( 1 \right)$
We know that the general equation is,
$a{x^2} + 2hxy + b{y^2} = 0$ ………………….equation$\left( 2 \right)$
On comparing equation $\left( 1 \right)$ and $\left( 2 \right)$ we get,
$a = 1,\;h = 2,\;b = 1$
Finding the angle between the lines,
$\tan \theta = \left| {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right|$
Substituting the values of a, h and b,
$\tan \theta = \left| {\dfrac{{2\sqrt {{{\left( {2} \right)}^2} - 1 \times 1} }}{{1 + 1}}} \right|$
$\tan \theta = \left| {\dfrac{{2\sqrt {4 - 1} }}{2}} \right|$
Solving it,
$\tan \theta = \left| {\dfrac{{2\sqrt 3 }}{2}} \right|$
$\tan \theta = \sqrt 3 $
Solving for $\theta ,$ we get,
$\theta = {60^ \circ }$
$\therefore $ The correct option is B.
Note: To solve this type of question one needs to be careful while writing the general equation and comparing the values of a, h and b. Avoid any type of calculation mistakes while substituting and solving the equation to get the correct angle.
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
