Minimum Value Of A² + B² Given 3A + 4B = 5

Jul 17, 2025 by ADMIN 43 views

Hey everyone! Today, we're diving into a super interesting math problem that combines algebra and geometry in a really neat way. We're going to figure out how to find the smallest possible value of A² + B² when we know that 3A + 4B = 5. This isn't just about crunching numbers; it's about understanding the relationship between equations and shapes. So, grab your thinking caps, and let's get started!

Understanding the Problem: Keywords to Success

Before we jump into the solution, let's break down what the question is really asking. The core of our problem lies in finding the minimum value. We are given the equation 3A + 4B = 5, which represents a straight line in a coordinate plane. We need to minimize the expression A² + B², which represents the square of the distance from the origin (0, 0) to a point (A, B). This means we're looking for the point on the line 3A + 4B = 5 that is closest to the origin. This understanding is crucial because it transforms an algebraic problem into a geometric one, making it easier to visualize and solve.

Keywords Breakdown and Why They Matter

Minimum Value: This immediately tells us we're not just solving for any value, but the smallest possible one. It implies there's a process of optimization involved.
A² + B²: Recognizing this as the square of the distance from the origin is key. It connects the algebra to a geometric concept.
3A + 4B = 5: This is our constraint. It limits the possible values of A and B to those that lie on this line. Understanding it's a line is vital.

By focusing on these keywords, we can translate the problem into a more intuitive form. We're essentially trying to find the shortest distance from a point (the origin) to a line. This geometric interpretation opens up different solution pathways, making the problem much more approachable.

Method 1: The Geometric Approach: Keywords in Action

Let's visualize this, guys. Imagine a line drawn on a graph representing 3A + 4B = 5. Now, picture circles of different sizes centered at the origin. The radius of each circle corresponds to the square root of A² + B². We're trying to find the smallest circle that just touches the line. That point of contact will give us the minimum value of A² + B².

Keywords Guiding the Geometric Solution

Shortest Distance: This is the heart of the geometric approach. The shortest distance from a point to a line is always along the perpendicular.
Perpendicular Distance: This is a crucial concept. The line segment connecting the origin to the point on the line with the minimum A² + B² will be perpendicular to 3A + 4B = 5.
Distance Formula: We'll use the formula for the distance from a point to a line to calculate this shortest distance.

Steps to the Geometric Solution

Visualize the Line: The equation 3A + 4B = 5 represents a straight line. We can rewrite it in slope-intercept form (B = (-3/4)A + 5/4) if that helps you visualize it better.
Shortest Distance is Perpendicular: The shortest distance from the origin to the line will be along a line perpendicular to 3A + 4B = 5.
Distance Formula Application: The distance (d) from a point (x₁, y₁) to a line Ax + By + C = 0 is given by the formula:
```
d = |Ax₁ + By₁ + C| / √(A² + B²)
```
In our case, (x₁, y₁) is the origin (0, 0), and our line is 3A + 4B - 5 = 0. So, A = 3, B = 4, and C = -5.

Calculate the Distance: Plugging the values into the formula, we get:

d = |(3 * 0) + (4 * 0) - 5| / √(3² + 4²)
d = |-5| / √(9 + 16)
d = 5 / √25
d = 5 / 5
d = 1

Distance and A² + B²: Remember, 'd' is the distance, which is equal to √(A² + B²). So, √(A² + B²) = 1.
Find A² + B²: Squaring both sides, we get A² + B² = 1² = 1.

Why This Works: Keywords and Their Power

The beauty of this approach is how it translates an algebraic problem into a geometric one. By recognizing that A² + B² is related to distance, we can use the concept of perpendicular distance to find the minimum value. The shortest distance is a key concept in geometry, and by applying the distance formula, we can precisely calculate this distance and, subsequently, the minimum value of A² + B².

Method 2: The Algebraic Approach: Keywords Unveiled

Now, let's tackle this problem using a purely algebraic method. This approach involves manipulating the given equation and expression to find the minimum value of A² + B² using a clever application of inequalities.

Keywords Guiding the Algebraic Solution

Cauchy-Schwarz Inequality: This is the star of our algebraic show! It provides a relationship between sums of squares and products.
Optimization: We're looking for the minimum value, so optimization techniques are relevant.
Relationship between Equations: We need to skillfully manipulate the given equation (3A + 4B = 5) and the expression we want to minimize (A² + B²).

Steps to the Algebraic Solution

Cauchy-Schwarz Inequality: The Cauchy-Schwarz Inequality states that for any real numbers a₁, a₂, b₁, b₂:
```
(a₁b₁ + a₂b₂)² ≤ (a₁² + a₂²) (b₁² + b₂²)
```
Applying the Inequality: Let's apply this to our problem. We can let a₁ = 3, a₂ = 4, b₁ = A, and b₂ = B. Plugging these into the inequality, we get:
```
(3A + 4B)² ≤ (3² + 4²) (A² + B²)
```
Substituting Known Values: We know that 3A + 4B = 5. Substituting this into the inequality, we have:
```
(5)² ≤ (9 + 16) (A² + B²)
25 ≤ 25 (A² + B²)
```
Solving for A² + B²: Dividing both sides by 25, we get:
```
1 ≤ A² + B²
```
Minimum Value: This tells us that the minimum value of A² + B² is 1.

Why This Works: Keywords and the Inequality

The magic of this method lies in the Cauchy-Schwarz Inequality. It provides a powerful tool for relating the sum of products to the product of sums of squares. By cleverly choosing our values (a₁, a₂, b₁, b₂), we can directly apply the inequality to our problem. The optimization aspect comes into play because the inequality gives us a lower bound for A² + B², which is exactly what we're looking for. This method demonstrates the elegance and power of algebraic manipulation in solving optimization problems.

Comparing the Methods: Keywords and Insights

Both the geometric and algebraic methods lead us to the same answer: the minimum value of A² + B² is 1. But each method offers a unique perspective and utilizes different key concepts.

Keywords for Comparison

Geometric Interpretation: The geometric approach provides a visual understanding of the problem, making it intuitive and easier to grasp for some.
Algebraic Manipulation: The algebraic approach relies on powerful inequalities and algebraic techniques, showcasing the elegance of mathematics.
Efficiency: Depending on your comfort level with geometry and algebra, one method might feel more efficient than the other.

Geometric vs. Algebraic: Which is Better?

Geometric: This method is great for visual learners. It uses the idea of the shortest distance and the distance formula to find the solution. It's very intuitive once you visualize the problem.
Algebraic: This method is more abstract but incredibly powerful. It uses the Cauchy-Schwarz Inequality, which is a fundamental tool in mathematics. It's efficient if you're comfortable with algebraic manipulations.

Ultimately, the