The number of real solutions (x, y, z, t) of simultaneous equations `2y=11/x+x` , ,` 2z=11/y+y` , `2t=11/z+z` , `2x=11/t+t` is

Options

A. 0
B. 1
C. 2
D. 4

Answer

To find the number of real solutions \((x, y, z, t)\) of the simultaneous equations:

1. \(2y = \frac{11}{x} + x\)
2. \(2z = \frac{11}{y} + y\)
3. \(2t = \frac{11}{z} + z\)
4. \(2x = \frac{11}{t} + t\)

we will convert each equation into a quadratic form and analyze the conditions for real solutions.

### Step 1: Convert the first equation into a quadratic form

Starting with the first equation:

\[ 2y = \frac{11}{x} + x \]

Multiply through by \(x\) (assuming \(x \neq 0\)) to eliminate the fraction:

\[ 2xy = 11 + x^2 \]

Rearranging gives:

\[ x^2 - 2xy + 11 = 0 \]

This is a quadratic equation in \(x\). The discriminant \(D\) must be non-negative for real solutions:

\[ D = b^2 - 4ac = (-2y)^2 - 4(1)(11) = 4y^2 - 44 \]

Setting the discriminant greater than or equal to zero:

\[ 4y^2 - 44 \geq 0 \]

Dividing by 4:

\[ y^2 \geq 11 \]

### Step 2: Convert the second equation into a quadratic form

Now for the second equation:

\[ 2z = \frac{11}{y} + y \]

Multiply through by \(y\):

\[ 2zy = 11 + y^2 \]

Rearranging gives:

\[ y^2 - 2zy + 11 = 0 \]

The discriminant for this equation is:

\[ D = (-2z)^2 - 4(1)(11) = 4z^2 - 44 \]

Setting the discriminant greater than or equal to zero:

\[ 4z^2 - 44 \geq 0 \]

Dividing by 4:

\[ z^2 \geq 11 \]

### Step 3: Convert the third equation into a quadratic form

For the third equation:

\[ 2t = \frac{11}{z} + z \]

Multiply through by \(z\):

\[ 2tz = 11 + z^2 \]

Rearranging gives:

\[ z^2 - 2tz + 11 = 0 \]

The discriminant is:

\[ D = (-2t)^2 - 4(1)(11) = 4t^2 - 44 \]

Setting the discriminant greater than or equal to zero:

\[ 4t^2 - 44 \geq 0 \]

Dividing by 4:

\[ t^2 \geq 11 \]

### Step 4: Convert the fourth equation into a quadratic form

For the fourth equation:

\[ 2x = \frac{11}{t} + t \]

Multiply through by \(t\):

\[ 2tx = 11 + t^2 \]

Rearranging gives:

\[ t^2 - 2tx + 11 = 0 \]

The discriminant is:

\[ D = (-2x)^2 - 4(1)(11) = 4x^2 - 44 \]

Setting the discriminant greater than or equal to zero:

\[ 4x^2 - 44 \geq 0 \]

Dividing by 4:

\[ x^2 \geq 11 \]

### Step 5: Analyze the results

From the above steps, we have derived the following conditions:

1. \(y^2 \geq 11\)
2. \(z^2 \geq 11\)
3. \(t^2 \geq 11\)
4. \(x^2 \geq 11\)

This implies:

- \(y \geq \sqrt{11}\) or \(y \leq -\sqrt{11}\)
- \(z \geq \sqrt{11}\) or \(z \leq -\sqrt{11}\)
- \(t \geq \sqrt{11}\) or \(t \leq -\sqrt{11}\)
- \(x \geq \sqrt{11}\) or \(x \leq -\sqrt{11}\)

### Step 6: Determine the number of solutions

Since \(x, y, z, t\) can either be all positive or all negative, we can have two cases:

1. All variables are positive: \(x = y = z = t = \sqrt{11}\)
2. All variables are negative: \(x = y = z = t = -\sqrt{11}\)

Thus, there are exactly **2 real solutions** for the system of equations.

### Final Answer

The number of real solutions \((x, y, z, t)\) is **2**.

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