a line L is such that its segment between the straight lines 5x - 4y = 4 and 3x +4y = 4 is bisected at the point (1,5). Find its equation

Let's solve the problem step by step.

Given:
1. The line \( L \) is bisected at the point \( (1, 5) \).
2. The segment is between two lines: \( 5x - 4y = 4 \) and \( 3x + 4y = 4 \).

### Step 1: Find the intersection points

To find the points where line \( L \) intersects the given lines, we need to find the points on these lines that have equal perpendicular distances from point \( (1, 5) \).

#### Finding perpendicular distances

The perpendicular distance \( d \) from a point \( (x_1, y_1) \) to a line \( ax + by + c = 0 \) is given by:

\[
d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}
\]

Let's calculate the perpendicular distances from the point \( (1, 5) \) to both lines.

1. **Distance from point \( (1, 5) \) to line \( 5x - 4y = 4 \):**

The line equation can be rewritten in the form \( ax + by + c = 0 \):

\[
5x - 4y - 4 = 0
\]

Here, \( a = 5 \), \( b = -4 \), and \( c = -4 \). The distance \( d_1 \) is:

\[
d_1 = \frac{|5(1) - 4(5) - 4|}{\sqrt{5^2 + (-4)^2}} = \frac{|5 - 20 - 4|}{\sqrt{25 + 16}} = \frac{|-19|}{\sqrt{41}} = \frac{19}{\sqrt{41}}
\]

2. **Distance from point \( (1, 5) \) to line \( 3x + 4y = 4 \):**

The line equation can be rewritten in the form \( ax + by + c = 0 \):

\[
3x + 4y - 4 = 0
\]

Here, \( a = 3 \), \( b = 4 \), and \( c = -4 \). The distance \( d_2 \) is:

\[
d_2 = \frac{|3(1) + 4(5) - 4|}{\sqrt{3^2 + 4^2}} = \frac{|3 + 20 - 4|}{\sqrt{9 + 16}} = \frac{|19|}{5} = \frac{19}{5}
\]

### Step 2: Find the equation of line \( L \)

Since line \( L \) is bisected at point \( (1, 5) \), it must pass through this point. Let the equation of line \( L \) be:

\[
y - 5 = m(x - 1)
\]

where \( m \) is the slope of line \( L \).

Now, we know that the line segment is bisected by line \( L \) at the point \( (1, 5) \). The perpendicular distance from the point \( (1, 5) \) to both lines \( 5x - 4y = 4 \) and \( 3x + 4y = 4 \) is equal. This implies that line \( L \) is the perpendicular bisector of the line segment joining the points of intersection of lines \( 5x - 4y = 4 \) and \( 3x + 4y = 4 \).

Let's denote these points of intersection as \( A(x_1, y_1) \) and \( B(x_2, y_2) \).

Now, we need to solve for the coordinates of points \( A \) and \( B \) by solving the system of equations.

### Step 3: Solve for the points of intersection

Let's solve for the coordinates of \( A \) and \( B \):

1. Solving for \( A \): Line \( 5x - 4y = 4 \)

Substitute the general form for \( y \):

2. Solving for \( B \): Line \( 3x + 4y = 4 \)

Now, solve for the equation of \( B \):

Then, we'll find the correct equation.

Learn how to find the equation of a line that bisects the segment between the lines 5x - 4y = 4 and 3x + 4y = 4 at the point (1, 5). This problem involves geometry concepts such as equations of lines and perpendicular bisectors. Suitable for high school students.

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