Question

A straight line through the point (4, 5) is such that its intercept between the axes is bisected at A, then its equation is

• A. 3x + 4y = 20
• B. 3x - 4y + 7 = 0
• C. 5x - 4y = 40
• D. 5x + 4y = 40

Hint:

If a point \( (h, k) \) bisects the intercept of a line between the coordinate axes, then the x-intercept is \( 2h \) and the y-intercept is \( 2k \). The equation of the line in intercept form is immediately \( \frac{x}{2h} + \frac{y}{2k} = 1 \).

Answer 1

The Correct Option is D

Let the equation of the line be in the intercept form: \( \frac{x}{a} + \frac{y}{b} = 1 \).
The x-intercept is the point where the line crosses the x-axis, which is \( (a, 0) \).
The y-intercept is the point where the line crosses the y-axis, which is \( (0, b) \).
The segment of the line between the axes has endpoints \( (a, 0) \) and \( (0, b) \).
We are given that this segment is bisected by the point A(4, 5). This means A is the midpoint of the segment.
Using the midpoint formula: \[ \left( \frac{a+0}{2}, \frac{0+b}{2} \right) = (4, 5) \]
This gives us two equations: \[ \frac{a}{2} = 4 \implies a = 8 \] \[ \frac{b}{2} = 5 \implies b = 10 \] Now we have the intercepts. Substitute these values back into the intercept form of the line's equation: \[ \frac{x}{8} + \frac{y}{10} = 1 \]
To convert this to the general form, find a common denominator (40) and multiply the entire equation by it: \[ 40 \left( \frac{x}{8} \right) + 40 \left( \frac{y}{10} \right) = 40 \times 1 \] \[ 5x + 4y = 40 \]
This matches option (D).