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Can anyone help me with this?

 

Find the largest value of x for which
x^2 + y^2 = 4x + 5y
has a solution, if x and y are real.

 Dec 17, 2023
 #1
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We can analyze the equation x^2 + y^2 = 4x + 5y in different ways to find the largest value of x for which it has a solution:

 

Method 1: Completing the square for x:

 

Move the x terms to one side: y^2 - 5y = x^2 - 4x

 

Complete the square for x by taking half of the coefficient of the x term, squaring it, and adding it to both sides: y^2 - 5y + 25/4 = (x^2 - 4x + 4) + 25/4 - 4

 

Rewrite the left side as a squared term: (y - 5/2)^2 = (x - 2)^2 - 1

 

Now, the equation represents a circle centered at (2, 5/2) with radius 1. Any real numbers (x, y) lying on or inside the circle satisfy the original equation.

 

Method 2: Using Cauchy-Schwarz Inequality:

 

Consider the equation in the form (x - 2)^2 = λ(y - 5/2)^2, where λ is a real number.

 

Apply Cauchy-Schwarz Inequality to (x - 2)^2 and λ(y - 5/2)^2: [(x - 2)(y - 5/2)]^2 <= (x - 2)^2 + λ(y - 5/2)^2: y - 5/2^2

 

Simplify: (xy - 5x/2 - 2y + 5) ^2 <= 0

 

Since the left side is a perfect square, it must be either equal to 0 or undefined. The latter case occurs when λ is negative, so we focus on λ >= 0.

 

Setting the left side to 0 and solving for x, we get: xy - 5x/2 - 2y + 5 = 0.

 

Finding the largest x:

 

The largest value of x corresponds to the case where the circle in Method 1 touches the positive x-axis. This happens when the radius is 0, meaning λ = 0 in Method 2. Substituting λ = 0 into the equation xy - 5x/2 - 2y + 5 = 0, we get: xy - 5x/2 - 2y + 5 = 0.

 

Solving for x using the quadratic formula gives:

x = (5 ± 5√5) / 4

 

Therefore, the largest value of x for which the equation has a solution is:

x = (5 + 5√5) / 4 ≈ 4.3027.

 

Note that this value lies on the positive x-axis, just touching the circle on the graph. Any larger value of x would result in a negative λ in Method 2, violating the Cauchy-Schwarz Inequality and leading to no real solutions for y.

 Dec 17, 2023

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