Find The Equation Of The Bisectors Of The Angle Between The Lines Given By $2x^2 - 7xy + 6y^2 = 0$.

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Introduction

In this article, we will explore the concept of finding the equation of the bisectors of the angle between two lines given by a quadratic equation in two variables. The quadratic equation in two variables represents a pair of lines, and the bisectors of the angle between these lines are the lines that divide the angle between the two given lines into two equal parts.

Understanding the Quadratic Equation

The given quadratic equation is $2x^2 - 7xy + 6y^2 = 0.$ This equation represents a pair of lines in the Cartesian plane. To find the equation of the bisectors of the angle between these lines, we need to first find the equations of the individual lines.

Finding the Equations of the Individual Lines

To find the equations of the individual lines, we can use the method of substitution or elimination. Let's use the method of substitution. We can rewrite the given equation as $2x^2 - 7xy + 6y^2 = (2x - 3y)(x - 2y) = 0.$ This gives us two equations: $2x - 3y = 0$ and $x - 2y = 0.$

Solving the First Equation

The first equation is $2x - 3y = 0.$ We can solve this equation for xx in terms of yy as follows:

2x=3y2x = 3y

x=3y2x = \frac{3y}{2}

Solving the Second Equation

The second equation is $x - 2y = 0.$ We can solve this equation for xx in terms of yy as follows:

x=2yx = 2y

Finding the Slope of the Lines

The slope of the first line is the coefficient of yy in the equation x=3y2x = \frac{3y}{2}. Therefore, the slope of the first line is 32\frac{3}{2}.

The slope of the second line is the coefficient of yy in the equation x=2yx = 2y. Therefore, the slope of the second line is 22.

Finding the Angle between the Lines

The angle between the two lines can be found using the formula:

tan⁑θ=∣m1βˆ’m21+m1m2∣\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|

where m1m_1 and m2m_2 are the slopes of the two lines.

Substituting the values of the slopes, we get:

tan⁑θ=∣32βˆ’21+32β‹…2∣\tan \theta = \left| \frac{\frac{3}{2} - 2}{1 + \frac{3}{2} \cdot 2} \right|

tan⁑θ=βˆ£βˆ’1272∣\tan \theta = \left| \frac{-\frac{1}{2}}{\frac{7}{2}} \right|

tan⁑θ=βˆ£βˆ’17∣\tan \theta = \left| -\frac{1}{7} \right|

ΞΈ=tanβ‘βˆ’1(17)\theta = \tan^{-1} \left( \frac{1}{7} \right)

Finding the Bisectors of the Angle

The bisectors of the angle between the two lines are the lines that divide the angle between the two given lines into two equal parts. The equations of the bisectors can be found using the formula:

y=m1+m22x+b1βˆ’b2m1βˆ’m2y = \frac{m_1 + m_2}{2} x + \frac{b_1 - b_2}{m_1 - m_2}

where m1m_1 and m2m_2 are the slopes of the two lines, and b1b_1 and b2b_2 are the y-intercepts of the two lines.

Since the two lines are given by the equations x=3y2x = \frac{3y}{2} and x=2yx = 2y, we can rewrite them in the slope-intercept form as follows:

y=23xy = \frac{2}{3} x

y=12xy = \frac{1}{2} x

The slopes of the two lines are 23\frac{2}{3} and 12\frac{1}{2}, respectively. The y-intercepts of the two lines are 00 and 00, respectively.

Substituting the values of the slopes and y-intercepts into the formula, we get:

y=23+122x+0βˆ’023βˆ’12y = \frac{\frac{2}{3} + \frac{1}{2}}{2} x + \frac{0 - 0}{\frac{2}{3} - \frac{1}{2}}

y=762xy = \frac{\frac{7}{6}}{2} x

y=712xy = \frac{7}{12} x

Conclusion

In this article, we have found the equation of the bisectors of the angle between the lines given by the quadratic equation $2x^2 - 7xy + 6y^2 = 0.$ The equation of the bisectors is $y = \frac{7}{12} x.$ This equation represents the lines that divide the angle between the two given lines into two equal parts.

References

  • [1] "Quadratic Equations in Two Variables" by [Author's Name], [Publisher's Name], [Year of Publication].
  • [2] "Bisectors of the Angle between Two Lines" by [Author's Name], [Publisher's Name], [Year of Publication].

Note: The references provided are fictional and for demonstration purposes only.

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Q1: What is the significance of finding the equation of the bisectors of the angle between the lines?

A1: Finding the equation of the bisectors of the angle between the lines is significant in various fields such as engineering, physics, and mathematics. It helps in understanding the properties of the lines and their relationships with each other.

Q2: How do I find the equation of the bisectors of the angle between the lines?

A2: To find the equation of the bisectors of the angle between the lines, you need to follow these steps:

  1. Find the equations of the individual lines using the given quadratic equation.
  2. Find the slopes of the individual lines.
  3. Find the angle between the two lines using the formula: $\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|$
  4. Find the bisectors of the angle using the formula: $y = \frac{m_1 + m_2}{2} x + \frac{b_1 - b_2}{m_1 - m_2}$

Q3: What is the formula for finding the angle between the two lines?

A3: The formula for finding the angle between the two lines is: $\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|$

Q4: What is the formula for finding the bisectors of the angle between the two lines?

A4: The formula for finding the bisectors of the angle between the two lines is: $y = \frac{m_1 + m_2}{2} x + \frac{b_1 - b_2}{m_1 - m_2}$

Q5: How do I find the slopes of the individual lines?

A5: To find the slopes of the individual lines, you need to rewrite the given quadratic equation in the slope-intercept form. The slope of the line is the coefficient of xx in the slope-intercept form.

Q6: What is the significance of the y-intercept in finding the equation of the bisectors of the angle between the lines?

A6: The y-intercept is not required to find the equation of the bisectors of the angle between the lines. However, it is required to find the equation of the individual lines.

Q7: Can I use any method to find the equation of the bisectors of the angle between the lines?

A7: Yes, you can use any method to find the equation of the bisectors of the angle between the lines. However, the method of substitution or elimination is recommended.

Q8: How do I verify the equation of the bisectors of the angle between the lines?

A8: To verify the equation of the bisectors of the angle between the lines, you need to substitute the values of xx and yy into the equation and check if it satisfies the given quadratic equation.

Q9: Can I find the equation of the bisectors of the angle between the lines using a calculator?

A9: Yes, you can find the equation of the bisectors of the angle between the lines using a calculator. However, it is recommended to use a graphing calculator or a computer algebra system to find the equation of the bisectors of the angle between the lines.

Q10: What are the applications of finding the equation of the bisectors of the angle between the lines?

A10: The applications of finding the equation of the bisectors of the angle between the lines are:

  • Engineering: Finding the equation of the bisectors of the angle between the lines is used in the design of bridges, buildings, and other structures.
  • Physics: Finding the equation of the bisectors of the angle between the lines is used in the study of optics and electromagnetism.
  • Mathematics: Finding the equation of the bisectors of the angle between the lines is used in the study of geometry and trigonometry.

Conclusion

In this article, we have answered some of the frequently asked questions about finding the equation of the bisectors of the angle between the lines. We have also provided the formulas and steps required to find the equation of the bisectors of the angle between the lines.