If `(x - ( 1)/( 3))^(2) + ( y - 4)^(2) = 0`, then what is the value of `( y +x)/( y -x)` ?

Options

A. `(11)/(13)`
B. ` ( 13)/( 11)`
C. `( 16)/( 9)`
D. `(9)/( 16)`

Answer

To solve the equation \((x - \frac{1}{3})^2 + (y - 4)^2 = 0\), we can follow these steps:

### Step 1: Analyze the equation
The equation is in the form of a sum of squares. The only way for this sum to equal zero is if both squares are individually equal to zero.

### Step 2: Set each square to zero
1. Set \((x - \frac{1}{3})^2 = 0\)
2. Set \((y - 4)^2 = 0\)

### Step 3: Solve for \(x\)
From \((x - \frac{1}{3})^2 = 0\):
\[ x - \frac{1}{3} = 0 \implies x = \frac{1}{3} \]

### Step 4: Solve for \(y\)
From \((y - 4)^2 = 0\):
\[ y - 4 = 0 \implies y = 4 \]

### Step 5: Substitute \(x\) and \(y\) into the expression \(\frac{y + x}{y - x}\)
Now, we have \(x = \frac{1}{3}\) and \(y = 4\). Substitute these values into the expression:
\[ \frac{y + x}{y - x} = \frac{4 + \frac{1}{3}}{4 - \frac{1}{3}} \]

### Step 6: Simplify the expression
1. Calculate the numerator:
\[ 4 + \frac{1}{3} = \frac{12}{3} + \frac{1}{3} = \frac{13}{3} \]
2. Calculate the denominator:
\[ 4 - \frac{1}{3} = \frac{12}{3} - \frac{1}{3} = \frac{11}{3} \]
3. Now substitute these into the fraction:
\[ \frac{\frac{13}{3}}{\frac{11}{3}} = \frac{13}{11} \]

### Final Answer
Thus, the value of \(\frac{y + x}{y - x}\) is \(\frac{13}{11}\).

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