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 the average of the coordinates of the two endpoints.

The given midpoint of the segment is \((1, 5)\).

2. The line passing through \((1, 5)\) is the locus of all points equidistant from the given lines

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

We use the formula for the distance of a point from a line:

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

Equate the distances of any point \((x, y)\) from the two lines:

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

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 the absolute values:

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

Expand both sides and simplify to find the equation of the locus:

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

Conclusion: The equation of the line is:

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