If a straight line passing through the point P(-3, 4) is such that its intercepted portion between t

Q: Wait, why do we assume the intercepts are $(a, 0)$ and $(0, b)$? Can't the line be vertical or horizontal?

That is a sharp observation! A vertical line would have an undefined slope and wouldn't intersect the y-axis (unless it is the y-axis itself). Since the problem specifies an 'intercepted portion between the coordinate axes', it implies the line must intersect both axes at finite, non-zero points. Thus, the intercept form is the most elegant and direct tool for this specific geometry.

Q: Why can't we just use the slope-point form $y - y_1 = m(x - x_1)$?

You absolutely can! However, you would first need to find the slope $m$. Since you know the intercepts are $(a, 0)$ and $(0, b)$, the slope is $m = \frac{b-0}{0-a} = -\frac{b}{a}$. It adds an extra layer of algebra. The intercept form $\frac{x}{a} + \frac{y}{b} = 1$ is designed specifically for this scenario, making it the 'path of least resistance'.

Q: Is there a shortcut for this in the exam?

Yes! If a point $P(x_1, y_1)$ bisects the intercepted portion of a line between the axes, the intercepts are always $a = 2x_1$ and $b = 2y_1$. It is a beautiful consequence of the midpoint formula. Just remember to use it only when the point is explicitly stated to be the midpoint!

Question

If a straight line passing through the point $P(-3, 4)$ is such that its intercepted portion between the coordinate axes is bisected at $P$, then its equation is :

Options:

(A) $x - y + 7 = 0$

(B) $3x - 4y + 25 = 0$

(C) $4x + 3y = 0$

(D) $4x - 3y + 24 = 0$

Sigma Percentile · Accepted Answer

### The Geometry of Balance: Unlocking the Intercepted Line

Welcome, future engineer. Today, we are not just solving an equation; we are exploring the elegant symmetry of the coordinate plane. Coordinate geometry is the bridge between the visual world of shapes and the rigorous world of algebra.

When you look at a problem like this, do not just see numbers. See a line, a dance between the x-axis and the y-axis, held in perfect balance by a single point.

### Phase 1: Visualizing the Space

Imagine standing on the Cartesian plane. We have a point $P(-3, 4)$. Because the x-coordinate is negative and the y-coordinate is positive, we are firmly in the second quadrant.

A line passes through this point, slicing through the axes. This line creates a segment—the 'intercepted portion'—that connects the x-axis at some point $A(a, 0)$ and the y-axis at some point $B(0, b)$. This segment is the stage upon which our problem unfolds.

### Phase 2: The Midpoint Logic

The problem gives us a gift: the point $P(-3, 4)$ bisects this segment $AB$. In the language of geometry, 'bisect' means to cut into two equal parts. This is our anchor.

If $P$ is the midpoint of the segment connecting $A(a, 0)$ and $B(0, b)$, then the coordinates of $P$ must be the average of the coordinates of $A$ and $B$. Mathematically, this is the midpoint formula:

$$P = \left( \frac{a + 0}{2}, \frac{0 + b}{2} ight)$$

This is where the magic happens. We equate this to our known point $P(-3, 4)$. This gives us two simple, beautiful equations:

$$\frac{a}{2} = -3 \quad 	ext{and} \quad \frac{b}{2} = 4$$

### Phase 3: The Algebraic Execution

Now, we solve for our intercepts. Multiplying by $2$ is trivial, but look at the result: $a = -6$ and $b = 8$. We have found the exact locations where our line kisses the axes.

The line crosses the x-axis at $-6$ and the y-axis at $8$. With these intercepts, we reach for the most powerful tool in our kit for this specific scenario: the Intercept Form of a line:

$$\frac{x}{a} + \frac{y}{b} = 1$$

Substituting our values, we get:

$$\frac{x}{-6} + \frac{y}{8} = 1$$

### Phase 4: The Final Polish

We are almost there. To make this look like the standard form found in textbooks and exams, we need to clear the fractions. The least common multiple of $6$ and $8$ is $24$.

Multiplying the entire equation by $24$, we get:

$$24 \left( \frac{x}{-6} ight) + 24 \left( \frac{y}{8} ight) = 24(1)$$

This simplifies to $-4x + 3y = 24$. Rearranging this to the standard form $Ax + By + C = 0$, we arrive at our final answer:

**$$4x - 3y + 24 = 0$$**

### The Pro-Tip: The Shortcut

As you prepare for the JEE Advanced, remember that speed is as important as accuracy. Whenever you see a problem where a point $P(x_1, y_1)$ bisects the intercepted portion, you can immediately write down the intercepts as $a = 2x_1$ and $b = 2y_1$.

It is a direct consequence of the midpoint formula, and it will save you precious seconds in the exam hall. Keep practicing, keep visualizing, and remember: every equation is just a story waiting to be told. You have the tools; now go out and master the geometry.

Animated Solution for Mathematics - Straight Lines: If a straight line passing through the point $P (- 3, 4)$ is such that its intercepted portion between the coordinate axes is bisected at $P$ , then its equation is :

Select Answer:

Visualized Solution

Visualizing the Point $P (- 3, 4)$

The Intercepted Portion

The Midpoint Condition

Applying the Midpoint Formula

Solving for the x-intercept $a$

Solving for the y-intercept $b$

The Intercept Form of a Line

Simplifying the Equation

Rearranging to Standard Form

Final Conclusion

The Sigma Insight: Various Forms of Equations of a Line

The Geometry of Balance

Phase 1

Phase 2

Phase 3

Phase 4

The Pro-Tip

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