A straight line L is perpendicular to the line 5x - y = 1. The area of the triangle formed by the li

Q: Wait, why do we need the absolute value in the area formula? Can't we just use $k$?

That is a fantastic question! In coordinate geometry, the intercepts $x = k$ and $y = k/5$ can be positive or negative depending on where the line cuts the axes. However, area is a physical quantity—it must always be positive. If we didn't use the absolute value, we might miss the solutions where $k$ is negative, which correspond to lines in the second or fourth quadrants. Always respect the geometry!

Q: Why are there two possible equations for the line $L$?

Think of the symmetry of the coordinate system. A line with a specific slope can be shifted parallel to itself and still maintain the same perpendicular relationship with the original line. There is one line on one side of the origin and another on the opposite side that both satisfy the area condition. Both are equally valid and mathematically beautiful!

Q: Can we use the intercept form $\frac{x}{a} + \frac{y}{b} = 1$ instead of $x + 5y = k$?

Absolutely! You can use $\frac{x}{a} + \frac{y}{b} = 1$. Since the slope is $-\frac{1}{5}$, you know that $-\frac{b}{a} = -\frac{1}{5}$, which means $a = 5b$. The area is $\frac{1}{2} |ab| = 5$. Substituting $a = 5b$, you get $\frac{1}{2} |5b^2| = 5$, so $b^2 = 2$. This leads to the same result. Both methods are powerful; choose the one that feels most intuitive to you!

Question

A straight line $L$ is perpendicular to the line $5x - y = 1$. The area of the triangle formed by the line $L$ and the coordinate axes is 5. Find the equation of the line $L$.

Sigma Percentile · Accepted Answer

### Analyzing the Reference Line

First, we must understand the soul of our reference line. By rearranging $5x - y = 1$ into the slope-intercept form, $y = 5x - 1$, we reveal its slope, $m_1 = 5$.

This tells us how steeply the line climbs. It is a line that rises five units for every one unit it moves to the right.

Now, for the perpendicular line $L$. The condition for perpendicularity is elegant and profound: the product of the slopes must be $-1$.

Thus, $m_1 \cdot m_2 = -1$, which leads us to the slope of our target line:

$$m_2 = -\frac{1}{5}$$

This negative reciprocal slope is the signature of perpendicularity.

### Defining the Family of Lines

With this slope, we define the family of lines $y = -\frac{1}{5}x + c$. This equation represents an infinite set of lines, all parallel to each other, all sharing the same slope.

To make our work cleaner, let us multiply the entire equation by 5 and rearrange it: $x + 5y = k$, where $k$ is a constant that determines the line's distance from the origin. This $k$ is the key that unlocks the specific line we are looking for.

### The Intercepts and the Area

Now, consider the intercepts. When $y = 0$, $x = k$. When $x = 0$, $y = \frac{k}{5}$.

These intercepts are the vertices of our triangle, sitting right on the axes. The area of a right-angled triangle is given by the formula:

$$	ext{Area} = \frac{1}{2} \cdot |x	ext{-intercept}| \cdot |y	ext{-intercept}|$$

Substituting our intercepts, we get the area as:

$$\frac{1}{2} |k| \cdot \left| \frac{k}{5} ight| = 5$$

This is the moment of truth. We are equating the geometric property of area to our algebraic constant $k$. Simplifying this, we find:

$$\frac{k^2}{10} = 5 \Rightarrow k^2 = 50$$

### The Final Revelation

Solving for $k$, we find $k = \pm 5\sqrt{2}$. The presence of the plus-or-minus sign is not a coincidence; it is a reflection of the symmetry of the coordinate system.

There are two lines, one on either side of the origin, that satisfy our condition. Substituting these values back into our general equation $x + 5y = k$, we arrive at our two final solutions:

**$x + 5y - 5\sqrt{2} = 0$** and **$x + 5y + 5\sqrt{2} = 0$**

We have successfully navigated the constraints, respected the geometry, and arrived at the truth. Remember, every equation tells a story. Today, the story was one of perpendicularity and area, and you have mastered it.

Animated Solution for Mathematics - Straight Lines: A straight line $L$ is perpendicular to the line $5 x - y = 1$ . The area of the triangle formed by the line $L$ and the coordinate axes is 5. Find the equation of the line $L$ .

Visualized Solution

Visualizing the Coordinate Plane

Slope of the Given Line

Slope of the Perpendicular Line $L$

General Equation of Line $L$

Locating the Intercepts

The Area Condition

Setting up the Area Equation

Simplifying the Equation

Solving for $k$

The Final Equations

The Sigma Insight: Various Forms of Equations of a Line

Analyzing the Reference Line

Defining the Family of Lines

The Intercepts and the Area

The Final Revelation

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