If `x^(2) + 9y^(2) = 1`, then minimum and maximum value of `3x^(2) - 27y^(2) + 24xy` respectively

Options

A. `0, 5`
B. `-7, 7`
C. `-5, 10`
D. `0, 10`

Explanation

To find the minimum and maximum values of the expression \( z = 3x^2 - 27y^2 + 24xy \) under the constraint \( x^2 + 9y^2 = 1 \), we can follow these steps:

### Step 1: Substitute Variables
We start with the constraint \( x^2 + 9y^2 = 1 \). We can express \( x \) and \( y \) in terms of a parameter \( \theta \):
- Let \( x = \cos \theta \)
- Then, substituting into the constraint gives us:
\[ \cos^2 \theta + 9y^2 = 1 \implies 9y^2 = 1 - \cos^2 \theta = \sin^2 \theta \]
- Therefore, we have:
\[ y^2 = \frac{1}{9} \sin^2 \theta \implies y = \frac{1}{3} \sin \theta \]

### Step 2: Substitute into the Expression
Now substitute \( x \) and \( y \) into the expression \( z \):
\[ z = 3(\cos^2 \theta) - 27\left(\frac{1}{3} \sin \theta\right)^2 + 24\left(\cos \theta\right)\left(\frac{1}{3} \sin \theta\right) \]
This simplifies to:
\[ z = 3\cos^2 \theta - 27 \cdot \frac{1}{9} \sin^2 \theta + 8 \sin \theta \cos \theta \]
\[ = 3\cos^2 \theta - 3\sin^2 \theta + 8\sin \theta \cos \theta \]

### Step 3: Use Trigonometric Identities
Using the identities:
- \( \cos^2 \theta - \sin^2 \theta = \cos 2\theta \)
- \( 2\sin \theta \cos \theta = \sin 2\theta \)

We can rewrite \( z \):
\[ z = 3\cos 2\theta + 4\sin 2\theta \]

### Step 4: Find Maximum and Minimum Values
The expression \( z = 3\cos 2\theta + 4\sin 2\theta \) can be analyzed using the amplitude formula:
- The maximum value of \( A\cos x + B\sin x \) is given by \( \sqrt{A^2 + B^2} \).
- Here, \( A = 3 \) and \( B = 4 \):
\[ \text{Maximum value} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
- The minimum value is the negative of the maximum value:
\[ \text{Minimum value} = -5 \]

### Final Result
Thus, the minimum and maximum values of \( z \) are:
- Minimum value: \(-5\)
- Maximum value: \(5\)