Understanding and Proving the Angle Bisectors of Two Given Lines
In this article, we will delve into the mathematical derivation of the angle bisectors of the given lines and then prove that these bisectors are at right angles to each other. The lines in question are:
Equations of the Given Lines
Line 1: (4x - 3y - 1 0) Line 2: (12x - 5y - 7 0)We will follow a systematic approach to find the equations of the bisectors and prove their orthogonality.
Step 1: Determine the Equations of the Lines
The given equations are already in the standard form:
Line 1:
(4x - 3y - 1 0)
Line 2:
(12x - 5y - 7 0)
Step 2: Find the Normal Forms of the Lines
The coefficients of the lines give us the direction vectors:
For Line 1:
(mathbf{n_1} 4 -3)
For Line 2:
(mathbf{n_2} 12 -5)
Step 3: Calculate the Angle Bisectors
The angle bisectors can be found using the formula:
[frac{L_1}{sqrt{A_1^2 B_1^2}} pm frac{L_2}{sqrt{A_2^2 B_2^2}}]where (L_1) and (L_2) are the equations of the lines in the form (Ax By C 0).
For the Given Lines:
(L_1 4x - 3y - 1), where (A_1 4), (B_1 -3), and (C_1 -1) (L_2 12x - 5y - 7), where (A_2 12), (B_2 -5), and (C_2 -7)Calculating the magnitudes:
For (L_1):
[sqrt{4^2 (-3)^2} sqrt{16 9} sqrt{25} 5]For (L_2):
[sqrt{12^2 (-5)^2} sqrt{144 25} sqrt{169} 13]Substituting these values into the bisector equation:
[frac{4x - 3y - 1}{5} pm frac{12x - 5y - 7}{13}]Step 4: Solve for the Angle Bisectors
We will solve for the bisectors in both the positive and negative cases.
Positive Case:
[frac{4x - 3y - 1}{5} frac{12x - 5y - 7}{13}]Cross-multiplying:
[begin{align*}13(4x - 3y - 1) 5(12x - 5y - 7)52x - 39y - 13 6 - 25y - 3552x - 39y - 13 - 6 25y 35 0-8x - 14y 22 04x 7y 11 quad text{(1)}end{align*}]Negative Case:
[frac{4x - 3y - 1}{5} -frac{12x - 5y - 7}{13}]Cross-multiplying:
[begin{align*}13(4x - 3y - 1) -5(12x - 5y - 7)52x - 39y - 13 -6 25y 3552x - 39y - 13 6 - 25y - 35 0112x - 64y - 48 07x - 4y -3 quad text{(2)}end{align*}]Step 5: Final Equations of the Bisectors
The equations of the angle bisectors are:
(4x 7y - 11 0) (7x - 4y - 3 0)Step 6: Proving that the Bisectors are at Right Angles
To show that the two bisectors are perpendicular, we need to find the slopes of the lines.
For (4x 7y - 11 0):
Rearranging gives:
[begin{align*}y -frac{4}{7}x - frac{11}{7}text{Slope} (m_1) -frac{4}{7}end{align*}]For (7x - 4y - 3 0):
Rearranging gives:
[begin{align*}y frac{7}{4}x - frac{3}{4}text{Slope} (m_2) frac{7}{4}end{align*}]To check if they are perpendicular, we can multiply the slopes:
[begin{align*}m_1 cdot m_2 left(-frac{4}{7}right) cdot left(frac{7}{4}right) -1end{align*}]Since the product of the slopes is (-1), the angle bisectors are indeed perpendicular to each other.
Conclusion
The equations of the angle bisectors are:
(4x 7y - 11 0) (7x - 4y - 3 0)These bisectors are at right angles to each other as proven by the product of their slopes being (-1).