Closest Point On A Line: A Step-by-Step Guide

Oct 20, 2025 by ADMIN 46 views

Finding the Closest Point on a Line: A Comprehensive Guide

Hey everyone! Today, we're diving into a classic problem in coordinate geometry: finding the point on a line that's closest to a given point. Specifically, we'll tackle the question: What is the point on the line 5x + 4y - 4 = 0 that is closest to the point (-4, -1)? This might sound intimidating, but don't worry, we'll break it down step-by-step so you can master this concept. So, grab your thinking caps, and let's get started!

Understanding the Problem: The Essence of Proximity

Before we jump into the calculations, let's make sure we understand what we're trying to achieve. Imagine you have a straight line stretching out infinitely in both directions. Then, picture a single point floating somewhere off that line. Our mission is to find the exact spot on the line that's nearest to that point. Think of it like you're standing at the point and trying to throw a ball to the line – you'd naturally aim for the closest spot, right?

In mathematical terms, the closest point is the one that minimizes the distance between the given point and any point on the line. This minimal distance is achieved along the line segment that is perpendicular to the given line and passes through the given point. This perpendicularity is the key to solving the problem. We're essentially looking for the foot of the perpendicular dropped from the point (-4, -1) onto the line 5x + 4y - 4 = 0. Now that we've visualized the problem, let's move on to the solution.

Why is this important?

This concept isn't just a theoretical exercise; it has practical applications in various fields. For instance, in computer graphics, it's used to determine the shortest distance between an object and a surface. In optimization problems, finding the closest point helps to minimize error or cost. So, understanding this principle can be incredibly useful in real-world scenarios.

Step-by-Step Solution: A Journey to the Closest Point

Okay, let's roll up our sleeves and solve this problem. We'll follow a step-by-step approach to make it super clear.

Step 1: Finding the Slope of the Given Line

Our first task is to figure out the slope of the line 5x + 4y - 4 = 0. To do this, we'll rewrite the equation in the slope-intercept form, which is y = mx + c, where 'm' represents the slope and 'c' is the y-intercept. Let's rearrange the equation:

5x + 4y - 4 = 0
4y = -5x + 4
y = (-5/4)x + 1

From this, we can see that the slope of the given line, which we'll call m1, is -5/4. Remember, the slope tells us how steep the line is and in which direction it's inclined.

Step 2: Determining the Slope of the Perpendicular Line

As we discussed earlier, the line segment connecting our point (-4, -1) to the closest point on the given line will be perpendicular to the given line. This is a crucial piece of information. We know that the slopes of two perpendicular lines are negative reciprocals of each other. In other words, if the slope of one line is m1, the slope of a line perpendicular to it, m2, will be -1/m1.

So, in our case:

m2 = -1 / m1
m2 = -1 / (-5/4)
m2 = 4/5

Therefore, the slope of the line perpendicular to 5x + 4y - 4 = 0 is 4/5. This new slope will help us define the equation of the line that passes through our given point and intersects the original line at the closest point.

Step 3: Finding the Equation of the Perpendicular Line

Now that we have the slope of the perpendicular line (4/5) and a point it passes through (-4, -1), we can use the point-slope form of a line equation to find its equation. The point-slope form is:

y - y1 = m(x - x1)

Where (x1, y1) is the given point and 'm' is the slope. Plugging in our values:

y - (-1) = (4/5)(x - (-4))
y + 1 = (4/5)(x + 4)

Let's simplify this to the slope-intercept form (y = mx + c):

y + 1 = (4/5)x + 16/5
y = (4/5)x + 16/5 - 1
y = (4/5)x + 11/5

So, the equation of the line perpendicular to 5x + 4y - 4 = 0 and passing through (-4, -1) is y = (4/5)x + 11/5. This is a significant step forward, as we now have equations for both the original line and the perpendicular line.

Step 4: Finding the Intersection Point

The point where the two lines intersect is the closest point we're searching for! To find this point, we need to solve the system of equations formed by the original line and the perpendicular line:

5x + 4y - 4 = 0  (Equation 1)
y = (4/5)x + 11/5  (Equation 2)

We can substitute Equation 2 into Equation 1 to solve for x:

5x + 4((4/5)x + 11/5) - 4 = 0
5x + (16/5)x + 44/5 - 4 = 0
(25x + 16x + 44 - 20) / 5 = 0
41x + 24 = 0
41x = -24
x = -24/41

Now that we have the x-coordinate, we can plug it back into either Equation 1 or Equation 2 to find the y-coordinate. Let's use Equation 2:

y = (4/5)(-24/41) + 11/5
y = -96/205 + 11/5
y = (-96 + 451) / 205
y = 355/205
y = 71/41

Therefore, the intersection point, which is the closest point on the line, is (-24/41, 71/41). We've successfully navigated through the calculations and pinpointed the exact location!

Step 5: Verifying the Solution (Optional but Recommended)

To be absolutely sure of our answer, it's always a good idea to verify it. One way to do this is to calculate the distance between the given point (-4, -1) and the point we found (-24/41, 71/41). Then, pick another point on the line 5x + 4y - 4 = 0 and calculate the distance between that point and (-4, -1). The distance to our closest point should be smaller.

Let's calculate the distance between (-4, -1) and (-24/41, 71/41) using the distance formula:

distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
distance = sqrt((-24/41 - (-4))^2 + (71/41 - (-1))^2)
distance = sqrt(((-24 + 164)/41)^2 + ((71 + 41)/41)^2)
distance = sqrt((140/41)^2 + (112/41)^2)
distance = sqrt((19600 + 12544) / 1681)
distance = sqrt(32144 / 1681)
distance ≈ 4.37

Now, let's pick another point on the line 5x + 4y - 4 = 0. If we set x = 0, we get 4y = 4, so y = 1. The point (0, 1) is on the line. Let's calculate the distance between (-4, -1) and (0, 1):

distance = sqrt((0 - (-4))^2 + (1 - (-1))^2)
distance = sqrt(4^2 + 2^2)
distance = sqrt(16 + 4)
distance = sqrt(20)
distance ≈ 4.47

As we can see, the distance to our calculated closest point (approximately 4.37) is indeed less than the distance to the arbitrarily chosen point (0, 1) (approximately 4.47). This gives us confidence that our solution is correct!

Conclusion: Mastering the Art of Proximity

There you have it! We've successfully found the point on the line 5x + 4y - 4 = 0 that is closest to the point (-4, -1). The closest point is (-24/41, 71/41). We achieved this by understanding the concept of perpendicularity, finding the slope of the perpendicular line, determining its equation, and then solving the system of equations to find the intersection point.

Remember, the key takeaways are:

The closest point lies on the line segment perpendicular to the given line.
The slopes of perpendicular lines are negative reciprocals of each other.
Solving a system of equations helps find the intersection point.

This problem showcases the beauty of coordinate geometry and its ability to solve geometric problems using algebraic techniques. By breaking down the problem into smaller steps, we made it manageable and understandable. Keep practicing, guys, and you'll become pros at these kinds of problems in no time! I hope this guide was helpful, and happy problem-solving!