Finding the Equation of a Straight Line Given Specific Intercepts and Midpoint Conditions

To find the equation of a straight line with intercepts in a specific ratio and that bisects a line segment, let’s follow a structured approach. This article will guide you through the steps, making use of the provided scenario where the intercepts are in the ratio 2:1 and the line bisects the segment joining the points (3, -4) and (5, 2).

Step-by-Step Solution

Step 1: Find the Midpoint of the Line Segment

The midpoint M of the line segment joining the points x_1, y_1 (3, -4) and x_2, y_2 (5, 2) can be calculated using the midpoint formula:

M left(frac{x_1 x_2}{2}, frac{y_1 y_2}{2}right)right)

Calculating this:

M left(frac{3 5}{2}, frac{-4 2}{2}right) (4, -1)

Step 2: Determine the Intercepts

Let the x-intercept be a and the y-intercept be b. According to the problem, the intercepts are in the ratio 2:1. Thus we can express the intercepts as:

a 2k and b k

for some constant k.

Step 3: Write the Equation of the Line

The equation of a line in terms of its intercepts a and b is given by:

frac{x}{a} frac{y}{b} 1

Substituting a 2k and b k:

frac{x}{2k} frac{y}{k} 1

Multiplying through by 2k:

x 2y 2k

Step 4: Find k Using the Midpoint

Since the line bisects the segment connecting the points (3, -4) and (5, 2), the midpoint (4, -1) must satisfy the line's equation:

4 2(-1) 2k

Calculating this gives:

4 - 2 2k quadRightarrowquad 2 2k quadRightarrowquad k 1

Step 5: Substitute k Back into the Intercepts

Now substituting k 1:

a 2(1) 2 and b 1

Step 6: Write the Final Equation

Substituting these values back into the line equation:

frac{x}{2} frac{y}{1} 1

Multiplying through by 2 to eliminate the fractions:

x 2y 2

Final Answer:

The equation of the straight line is:

boxed{x 2y 2}

Additional Solutions

We can verify this solution by considering another line with intercepts 2a and a units:

Solution 1

The equations of the lines are:

frac{x}{2a} frac{y}{a} 1quadtext{and}quadfrac{x}{a} frac{y}{2a} 1

Line 1 passes through the midpoint M of (3, -4) and (5, 2), which is (4, -1).

Substituting the midpoint (4, -1) into the equation:

frac{4}{2a} frac{-1}{a} 1

This simplifies to:

frac{2}{a} - frac{1}{a} 1

frac{1}{a} 1 quadRightarrowquad a 1

Substituting a 1 into the first equation:

frac{x}{2(1)} frac{y}{1} 1

This simplifies to:

x 2y 2

Line 2 also passes through the midpoint M (4, -1).

Substituting the midpoint (4, -1) into the equation:

frac{4}{a} frac{-1}{2a} 1

This simplifies to:

frac{7}{2a} 1 quadRightarrowquad a frac{7}{2}

Substituting a frac{7}{2} into the second equation:

frac{2x}{7} frac{y}{7} 1

This simplifies to:

2x y 7

Therefore, there are two possible equations for the line, but the specific equation required by the problem is:

boxed{x 2y 2}