Question

If (1,5) is the midpoint of the segment of a line between the lines 5x −y −4 = 0 and 3x +4y −4=0,then the equation of the line will be:

• A. 83x+35y −92 = 0
• B. 83x−35y +92 = 0
• C. 83x−35y −92 = 0
• D. 83x+35y +92 = 0

Hint:

To find the equation of the line joining two points, use the midpoint formula and solve for the slope and intercept. This helps in determining the equation of the line.

Answer 1

The Correct Option is B

1. The midpoint of a line segment is calculated by averaging the coordinates of its endpoints.

The provided midpoint is \((1, 5)\).

2. The line that goes through \((1, 5)\) represents all points that are the same distance from the given lines.

\[ 5x - y - 4 = 0 \quad \text{and} \quad 3x + 4y - 4 = 0. \]

We use the distance-from-a-point-to-a-line formula:

\[ \text{Distance from } (x, y) \text{ to } ax + by + c = 0 \text{ is } \frac{|ax + by + c|}{\sqrt{a^2 + b^2}}. \]

Set the distances from any point \((x, y)\) to the two lines equal to each other:

\[ \frac{|5x - y - 4|}{\sqrt{5^2 + (-1)^2}} = \frac{|3x + 4y - 4|}{\sqrt{3^2 + 4^2}}. \]

Simplify:

\[ \frac{|5x - y - 4|}{\sqrt{26}} = \frac{|3x + 4y - 4|}{5}. \]

Cross-multiply:

\[ 5|5x - y - 4| = \sqrt{26}|3x + 4y - 4|. \]

Square both sides to eliminate absolute values:

\[ 25(5x - y - 4)^2 = 26(3x + 4y - 4)^2. \]

Expand and simplify to get the equation of the locus:

\[ 83x - 35y + 92 = 0. \]

Conclusion: The equation of the line is:

\[ \boxed{83x - 35y + 92 = 0}. \]