Question

A straight line through the point \( A(3, 4) \) is such that its intercept between the axes is bisected at \( A \). Its equation is:

• A. \( 3x - 4y + 7 = 0 \)
• B. \( 4x + 3y = 24 \)
• C. \( 3x + 4y = 25 \)
• D. \( x + y = 7 \)

Hint:

To find the equation of a line whose intercepts are bisected at a point, use the midpoint formula for the intercepts and substitute the values into the general intercept form of the line.

Answer 1

The Correct Option is B

We are given that a straight line passes through the point \( A(3, 4) \), and the intercepts between the axes are bisected at point \( A \). We need to find the equation of this line. Step 1: General form of the line.
The equation of a straight line in terms of the intercepts on the axes is given by: \[ \frac{x}{a} + \frac{y}{b} = 1, \] where \( a \) is the x-intercept and \( b \) is the y-intercept.
Step 2: Condition of bisection.
We are told that the line’s intercepts are bisected at point \( A(3, 4) \). This means the midpoint of the intercepts must be \( A \). The x-intercept of the line is \( a \), and the y-intercept is \( b \). The midpoint of the intercepts is given by: \[ \left( \frac{a}{2}, \frac{b}{2} \right) = (3, 4). \] This gives the equations: \[ \frac{a}{2} = 3 \quad \Rightarrow \quad a = 6, \] \[ \frac{b}{2} = 4 \quad \Rightarrow \quad b = 8. \]
Step 3: Writing the equation of the line.
Now that we know the intercepts \( a = 6 \) and \( b = 8 \), we can write the equation of the line: \[ \frac{x}{6} + \frac{y}{8} = 1. \] Multiply through by 24 (the least common multiple of 6 and 8) to clear the denominators: \[ 4x + 3y = 24. \]
Step 4: Conclusion.
Therefore, the equation of the line is \( 4x + 3y = 24 \), and the correct answer is (b).