Question

The equation of line containing the angle bisector of the lines \(3 x-4 y-2=0\) and \(5 x-12 y+2=0\) is ....... (a) \(7 x+4 y-18=0\) (b) \(4 x-7 y-1=0\) (c) \(4 \mathrm{x}-7 \mathrm{y}+1=0\) (d) None of these

Short answer

(b) \(4x - 7y - 1 = 0\)

Step-by-step solution

Find the slopes of the given lines

To find the slopes of the lines, rewrite both of them in the slope-intercept form, y = mx + b, where m is the slope. Original equations: \(3x - 4y - 2 = 0\) \(5x - 12y + 2 = 0\) Lines in slope-intercept form: \(y = \frac{3}{4}x - \frac{1}{2}\) (Line 1) \(y = \frac{5}{12}x - \frac{1}{6}\) (Line 2) Slopes: \(m_1 = \frac{3}{4}\) \(m_2 = \frac{5}{12}\)

Determine the tangent of the half-angle using the slopes

Now, we will find the tangent of the half-angle, using the following formula: \(tan(\frac{\phi}{2}) = \frac{m_2 - m_1}{1 + m_1 m_2}\) Plug in the slopes from Step 1: \(tan(\frac{\phi}{2}) = \frac{\frac{5}{12} - \frac{3}{4}}{1 + (\frac{3}{4})(\frac{5}{12})}\) Simplify the expression: \(tan(\frac{\phi}{2}) = \frac{\frac{-1}{3}}{\frac{25}{16}} = -\frac{16}{75}\) Since the tangent of the half-angle is negative, we know that one of the angle bisectors will have a positive slope, and the other will have a negative slope. At this point, we can observe which of the given answer choices have lines with slopes of opposite signs. (a) 7x + 4y - 18 = 0 → y = -\(\frac{7}{4}\)x + \(\frac{9}{2}\) → negative slope (b) 4x - 7y - 1 = 0 → y = \(\frac{4}{7}\)x - \(\frac{1}{7}\) → positive slope (c) 4x - 7y + 1 = 0 → y = \(\frac{4}{7}\)x + \(\frac{1}{7}\) → positive slope From the elimination above, we can conclude that the angle bisector lies between lines (b) and (c). To determine which one is correct, observe the numerator of the tangent expression and match the signs to the slopes of the given lines.

Match the signs with the denominator of the tangent expression

To match the signs, compare the signs of the differences in the slopes of the given lines and the tangent expression (numerator). Since our tangent of the half-angle is negative, we are looking for a pair of lines where the difference in their slopes is negative. \(m_b = \frac{4}{7}\), \(m_c = \frac{4}{7}\) (slope of lines (b) and (c)) \(m_1 = \frac{3}{4}\) (slope of Line 1) Differences: \(m_b - m_1 = \frac{4}{7} - \frac{3}{4} = \frac{1}{28}\) → positive \(m_c - m_1 = \frac{4}{7} - \frac{3}{4} = \frac{1}{28}\) → positive Since both differences are positive, the angle bisector must be between Line 1 and the reflection of either (b) or (c). The reflection of a line with a positive slope will make the slope negative. Reflect the given line equations (b) and (c): Reflected lines: (b') y = -\(\frac{4}{7}\)x + \(r_1\) (c') y = -\(\frac{4}{7}\)x + \(r_2\) Now, let's calculate the differences again between Line 1 and the reflected lines (b') and (c'): Differences: \(m_{b'} - m_1 = -\frac{4}{7} - \frac{3}{4} = -\frac{25}{28}\) → negative \(m_{c'} - m_1 = -\frac{4}{7} - \frac{3}{4} = -\frac{25}{28}\) → negative Now both differences are negative, and the angle bisector must be between Line 1 and the reflection of either (b') or (c'). Note that -\(\frac{25}{28}\) is a negative version of the denominator from the tangent expression so it represents the correct angle bisector. From this, we can conclude that the equation of the line containing the angle bisector is:

Answer

(b) \(4x - 7y - 1 = 0\)

Key concepts

Slopes of Lines

Understanding the slope of a line is crucial for analyzing and graphing linear equations. It tells us how steep the line is and in which direction it goes.
To determine the slope, we first need to write a line's equation in the slope-intercept form, which is given by the formula \(y = mx + b\). Here, \(m\) is the slope, and \(b\) is the y-intercept, which indicates where the line crosses the y-axis.

• Equation 1: \(3x - 4y - 2 = 0\) can be rearranged to \(y = \frac{3}{4}x - \frac{1}{2}\), giving a slope \(m_1 = \frac{3}{4}\).
• Equation 2: \(5x - 12y + 2 = 0\) can be rearranged to \(y = \frac{5}{12}x - \frac{1}{6}\), giving a slope \(m_2 = \frac{5}{12}\).

The differences in these slopes show us how lines compare in steepness and direction, which is foundational when finding the angle bisector of these lines.

Tangent of Half-Angle

The tangent of the half-angle formula is a powerful tool in understanding the angle bisector of two lines. When two lines intersect, they form an angle with two angle bisectors—one internal and one external. The tangent of the half-angle provides a mathematical method for determining the slope of these bisectors.

To use the tangent of the half-angle formula, you'll need the slopes of the two lines involved. The formula is:
\[\tan\left(\frac{\phi}{2}\right) = \frac{m_2 - m_1}{1 + m_1 m_2}\]
Plug in the slopes from earlier: \(\tan\left(\frac{\phi}{2}\right) = \frac{\frac{5}{12} - \frac{3}{4}}{1 + (\frac{3}{4})(\frac{5}{12})}\)
Simplifying this, we find \(\tan\left(\frac{\phi}{2}\right) = -\frac{16}{75}\). This negative result indicates the bisector's angle is oriented such that one segment has a positive slope, and the other is negative.
Understanding this formula helps identify where each angle bisector lies, guiding us to the correct equation of the line that represents the bisector.

Equation of a Line

The final step in addressing the problem requires identifying the correct equation of a line containing the angle bisector. Once we know the tangent of the half-angle, it can guide us towards the right answer if we compare it with potential line equations.
Among the options, we're hunting for a line equation where the slope and the tangent's characteristics align.

• Equation (b): \(4x - 7y - 1 = 0\) results in \(y = \frac{4}{7}x - \frac{1}{7}\), which represents a positive slope.

Considering the characteristics of the tangent of the half-angle, reflecting the line equations provides options where the correct line having a negative slope can be considered:

• The reflected equation of (b) results in a negative slope, \(-\frac{4}{7}\).

When aligning this with the tangent's negative slope, the equation belonging to the angle bisector is identified as Equation (b). This approach offers clarity in selecting the right bisector line.