Q.If the coordinates of the middle point of the portion of a line intercepted between the coordinate axes is $(3,2)$, then the equation of the line will be
(A) $2x+3y=12$
(B) $3x+2y=12$
(C) $4x-3y=6$
(D) $5x-2y=10$

The key idea here is using the intercept form of a line and the midpoint formula.

Let the line intersect the x-axis at $(a,0)$ and the y-axis at $(0,b)$.

The midpoint of the segment intercepted between the axes is $\left(\frac{a+0}{2}, \frac{0+b}{2}\right) = \left(\frac{a}{2}, \frac{b}{2}\right)$.

Given that the midpoint is $(3,2)$, we have:

$\frac{a}{2} = 3 \implies a=6$

$\frac{b}{2} = 2 \implies b=4$

The equation of a line in intercept form is $\frac{x}{a} + \frac{y}{b} = 1$.

Substituting $a=6$ and $b=4$:

$\frac{x}{6} + \frac{y}{4} = 1$ …

When a line is "intercepted between the coordinate axes," it means we are considering the segment of the line that lies in the first quadrant (or generally, between the points where it crosses the x-axis and the y-axis). This segment connects the x-intercept and the y-intercept. The problem provides the midpoint of this specific segment.

The most natural way to represent a line when its intercepts are involved is the intercept form. This form directly uses the points where the line crosses the axes.

The problem states that the given point $(3,2)$ is the midpoint of the segment connecting these two intercept points. We can use the midpoint formula to relate $a$, $b$, and the given midpoint coordinates.

Here's how we solve it step-by-step:

1. Identify the intercept points:

   Let the equation of the line be $\frac{x}{a} + \frac{y}{b} = 1$.

   This line intersects the x-axis at point $A = (a, 0)$ and the y-axis at point $B = (0, b)$.

   The "portion of a line intercepted between the coordinate axes" is the line segment $AB$.

2. Apply the midpoint formula:

   The coordinates of the midpoint $M$ of a segment connecting $(x_1, y_1)$ and $(x_2, y_2)$ are given by $M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$.

   In our case, the two points are $A(a, 0)$ and $B(0, b)$, and the midpoint is given as $(3, 2)$.

   So, we have:

   $3 = \frac{a+0}{2}$

   $2 = \frac{0+b}{2}$

3. Solve for the intercepts $a$ and $b$: …