Animated Solution for Mathematics - Straight Lines: A straight line through the point $A(3,4)$ is such that its intercept between the axes is bisected at $A$. Its equation is

Imagine you are standing on a coordinate plane with a point $A(3,4)$ floating in the first quadrant. A line passes through this point, acting as a bridge that connects the $x$-axis and the $y$-axis.

The problem states that the segment of this line trapped between the axes is perfectly bisected at point $A$. This symmetry provides the key to unlocking the equation of the line.

When the line intersects the $x$-axis, its $y$-coordinate is zero. Let us define this point as $P(a,0)$.

Similarly, when the line intersects the $y$-axis, its $x$-coordinate is zero. Let us define this point as $Q(0,b)$.

The segment $PQ$ is the portion of the line trapped between the axes. We are given that $A(3,4)$ is the midpoint of $PQ$, meaning $A$ is the average of the coordinates of $P$ and $Q$.

The midpoint formula is our most reliable tool here. It states that the midpoint of a segment with endpoints $(x_1,y_1)$ and $(x_2,y_2)$ is:

Applying this to our points $P(a,0)$ and $Q(0,b)$, we obtain the midpoint:

Since this must equal $(3,4)$, we set up two simple equations:

Solving these yields $a=6$ and $b=8$. We have successfully identified the location of our intercepts.

Now that we have the intercepts $a=6$ and $b=8$, we can utilize the elegant intercept form of a line:

Substituting our values, we get:

To simplify this expression, we multiply the entire equation by the least common multiple of $6$ and $8$, which is $24$:

This simplifies beautifully to the final equation:

The Takeaway

We have arrived at our answer: $4x+3y=24$. The beauty of this problem lies in the realization that the midpoint of an intercept segment is always related to the intercepts by a factor of two.

If you encounter this on a JEE paper, remember the shortcut: $a=2x_1$ and $b=2y_1$. This powerful observation turns a complex-looking problem into a simple, rapid calculation.