Distance From Origin To Line $3x - Y + 1 = 0$

Ever wondered how far a specific point is from a line, especially that special point, the origin (0,0)? It's a fundamental concept in geometry and has practical applications in various fields, from computer graphics to engineering. Today, we're diving deep into how to find the distance from the origin to the graph of the equation $3x - y + 1 = 0$. This isn't just about crunching numbers; it's about understanding the spatial relationship between a point and a line. We'll break down the formula, walk through the steps, and even touch upon why this concept is so important. So, grab your thinking cap, and let's embark on this mathematical journey to find that precise distance. We'll explore the elegance of the distance formula and how it elegantly resolves this geometric puzzle. Understanding this relationship helps visualize geometric scenarios, making complex problems more approachable and solvable.

Unpacking the Distance Formula: Your Geometric Toolkit

Before we can calculate the distance from the origin to the line $3x - y + 1 = 0$, we need to equip ourselves with the right tools. The primary tool here is the formula for the distance between a point $(x_0, y_0)$ and a line $Ax + By + C = 0$. This formula is a cornerstone of coordinate geometry and provides a direct route to our answer. It's expressed as:

d = rac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}

Let's break this down. The numerator, $|Ax_0 + By_0 + C|$, represents the absolute value of plugging the coordinates of our point into the line's equation. The absolute value is crucial because distance is always a non-negative quantity. The denominator, $\sqrt{A^2 + B^2}$, is derived from the coefficients of the $x$ and $y$ terms in the line's equation. It essentially normalizes the equation, ensuring we get the shortest, perpendicular distance.

In our specific problem, the point we're interested in is the origin, which has coordinates $(x_0, y_0) = (0, 0)$. The line is given by the equation $3x - y + 1 = 0$. To use the formula, we need to identify the coefficients $A$, $B$, and $C$. By comparing $3x - y + 1 = 0$ to the general form $Ax + By + C = 0$, we can see that:

• $A = 3$
• $B = -1$
• $C = 1$

With these values identified, we're ready to plug them into the distance formula and solve for the distance.

Calculating the Distance: Step-by-Step

Now that we have our formula and our values, let's put them to work to find the distance from the origin to the line $3x - y + 1 = 0$. We'll substitute $x_0 = 0$, $y_0 = 0$, $A = 3$, $B = -1$, and $C = 1$ into the distance formula:

d = rac{|(3)(0) + (-1)(0) + 1|}{\sqrt{(3)^2 + (-1)^2}}

Let's simplify the numerator first. Since any number multiplied by zero is zero, $(3)(0) = 0$ and $(-1)(0) = 0$. So, the numerator becomes $|0 + 0 + 1|$, which simplifies to $|1|$. The absolute value of 1 is simply 1.

Now, let's simplify the denominator. We have $(3)^2 = 9$ and $(-1)^2 = 1$. So, the expression under the square root is $9 + 1 = 10$. The denominator is then $\sqrt{10}$.

Putting it all together, our distance $d$ is:

d = rac{1}{\sqrt{10}}

This is a valid answer, but often, we prefer to rationalize the denominator. To do this, we multiply both the numerator and the denominator by $\sqrt{10}$:

d = rac{1}{\sqrt{10}} imes rac{\sqrt{10}}{\sqrt{10}} = rac{\sqrt{10}}{10}

So, the distance from the origin to the line $3x - y + 1 = 0$ is $\frac{\sqrt{10}}{10}$. This is a precise, numerical answer that tells us exactly how far the origin is from that specific line.

Visualizing the Solution: A Geometric Interpretation

It's always helpful to visualize what we've just calculated. The distance from the origin to the line $3x - y + 1 = 0$ represents the length of the shortest line segment that can be drawn from the origin to any point on the line. This shortest segment is always perpendicular to the line itself. Imagine the line $3x - y + 1 = 0$ drawn on a graph. The origin (0,0) is a fixed point. The distance we found, $\frac{\sqrt{10}}{10}$, is the length of the perpendicular from (0,0) to that line.

Consider the line's properties. The equation $3x - y + 1 = 0$ can be rewritten in slope-intercept form ($y = mx + b$) as $y = 3x + 1$. This tells us the line has a y-intercept of 1 (it crosses the y-axis at the point (0,1)) and a slope of 3. The slope of 3 means that for every 1 unit we move to the right on the x-axis, we move 3 units up on the y-axis.

The perpendicular line from the origin to this line will have a slope that is the negative reciprocal of 3, which is $-\frac{1}{3}$. We could, in theory, find the equation of this perpendicular line passing through the origin (which would be $y = -\frac{1}{3}x$) and then find the intersection point of this perpendicular line with our original line $y = 3x + 1$. Once we have that intersection point $(x_i, y_i)$, we could use the standard distance formula between two points, $(0,0)$ and $(x_i, y_i)$, to find the distance. This would be $\sqrt{(x_i - 0)^2 + (y_i - 0)^2} = \sqrt{x_i^2 + y_i^2}$.

While this method is more involved, it confirms the geometric interpretation. The formula we used bypasses this multi-step process by directly calculating the shortest distance. The value $\frac{\sqrt{10}}{10}$ is a small, positive number, which intuitively makes sense given the line's position relative to the origin. It's not excessively far, but it's certainly not zero.

Why Does This Matter? Applications and Significance

Understanding how to calculate the distance from the origin to a line is more than just an academic exercise. This concept has tangible applications across various domains. In computer graphics, for instance, determining the distance from a light source (often treated as the origin) to a surface (represented by planes or lines) is crucial for calculating lighting effects and shadows. In robotics, it can be used to ensure a robot arm doesn't collide with obstacles or to calculate the proximity of a target.

In physics, concepts of potential energy or fields often relate to distances from a central point or source. For example, gravitational or electric fields decrease in strength with distance from their source. While these are often in three dimensions, the 2D principle of distance from a point to a line is foundational.

Furthermore, in optimization problems within mathematics and engineering, finding the minimum distance from a set of constraints (lines or planes) to a desired state (a point) is a common task. The formula provides an efficient way to quantify this relationship. It's a building block for more complex calculations involving geometry and spatial relationships. Without this fundamental understanding, many advanced mathematical and scientific models would be inaccessible. The elegance of the formula lies in its ability to distill a complex geometric relationship into a simple algebraic expression, making it a powerful tool in the hands of mathematicians, scientists, and engineers alike.

Conclusion: The Precise Answer

We set out to find the distance from the origin to the graph of the equation $3x - y + 1 = 0$. By employing the standard formula for the distance between a point and a line, and carefully substituting our values, we arrived at a clear and precise answer. The origin is $(0,0)$, and the line is $3x - y + 1 = 0$. Applying the formula d = rac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}, we found:

• $A = 3$, $B = -1$, $C = 1$
• $x_0 = 0$, $y_0 = 0$
• Numerator: $|(3)(0) + (-1)(0) + 1| = |1| = 1$
• Denominator: $\sqrt{(3)^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10}$

This gives us $d = \frac{1}{\sqrt{10}}$, which, after rationalizing the denominator, becomes $d = \frac{\sqrt{10}}{10}$.

Comparing this to the given options:

a. 10 b. $\sqrt{10}$ c. $\frac{\sqrt{10}}{10}$ d. $10 \sqrt{10}$

Our calculated distance matches option c. This mathematical journey highlights how a straightforward formula can solve a seemingly complex geometric problem, providing a concrete measure of spatial separation.

For further exploration into the fascinating world of coordinate geometry and distance calculations, you might find these resources helpful:

• Khan Academy: Distance between a point and a line
• MathWorld: Distance from a Point to a Line