Find three largest non-zero values represented by the form −x2+5xy−11y2−x2+5xy−11y2.

(−1,−3,−5)(−1,−3,−5)

(−1,−5,−7)(−1,−5,−7)

(−3,−5,−7)(−3,−5,−7)

(−3,−7,−9)(−3,−7,−9)

(−5,−7,−9)(−5,−7,−9)

(−5,−8,−9)

given a multiple selection question, give answer please.

Let's solve this problem step by step. We are given the quadratic form:

\[
f(x, y) = -x^2 + 5xy - 11y^2
\]

We need to find the three largest non-zero values for this expression, and we're given several points \((x, y)\) to evaluate.

### Step 1: Evaluate \(f(x, y)\) for each given pair.

1. **For \((-1, -3)\):**

\[
f(-1, -3) = -(-1)^2 + 5(-1)(-3) - 11(-3)^2
\]
\[
= -1 + 15 - 11(9) = -1 + 15 - 99 = -85
\]

2. **For \((-1, -5)\):**

\[
f(-1, -5) = -(-1)^2 + 5(-1)(-5) - 11(-5)^2
\]
\[
= -1 + 25 - 11(25) = -1 + 25 - 275 = -251
\]

3. **For \((-3, -5)\):**

\[
f(-3, -5) = -(-3)^2 + 5(-3)(-5) - 11(-5)^2
\]
\[
= -9 + 75 - 11(25) = -9 + 75 - 275 = -209
\]

4. **For \((-3, -7)\):**

\[
f(-3, -7) = -(-3)^2 + 5(-3)(-7) - 11(-7)^2
\]
\[
= -9 + 105 - 11(49) = -9 + 105 - 539 = -443
\]

5. **For \((-5, -7)\):**

\[
f(-5, -7) = -(-5)^2 + 5(-5)(-7) - 11(-7)^2
\]
\[
= -25 + 175 - 11(49) = -25 + 175 - 539 = -389
\]

6. **For \((-5, -8)\):**

\[
f(-5, -8) = -(-5)^2 + 5(-5)(-8) - 11(-8)^2
\]
\[
= -25 + 200 - 11(64) = -25 + 200 - 704 = -529
\]

### Step 2: Compare the values.

The values are:
- \((-1, -3)\): \(-85\)
- \((-1, -5)\): \(-251\)
- \((-3, -5)\): \(-209\)
- \((-3, -7)\): \(-443\)
- \((-5, -7)\): \(-389\)
- \((-5, -8)\): \(-529\)

### Step 3: Identify the three largest values.

The largest values are:
- \(-85\)
- \(-209\)
- \(-251\)

### Answer:

The points that give the three largest non-zero values are:
- \((-1, -3)\)
- \((-3, -5)\)
- \((-1, -5)\)

These correspond to the second option: **\((-1, -3), (-1, -5), (-3, -5)\)**.

Do you want further details or have any questions?

### 5 Related Questions:
1. How do quadratic forms like this relate to conic sections?
2. What are the properties of the quadratic function \(f(x, y) = -x^2 + 5xy - 11y^2\)?
3. How do you determine the maximum or minimum values of a quadratic function?
4. Can this form \(f(x, y)\) be rewritten using matrix notation?
5. How can we visualize the function \(f(x, y)\) geometrically?

### Tip:
Always double-check each substitution and arithmetic step carefully to avoid small mistakes, especially when dealing with negative signs!

Find three largest non-zero values represented by the form −x^2 + 5xy − 11y^2 for the points (-1,-3), (-1,-5), (-3,-5), (-3,-7), (-5,-7), (-5,-8).

Explore how to find the three largest non-zero values represented by the quadratic form −x^2 + 5xy − 11y^2 by evaluating given points. This problem involves algebraic functions of two variables, ideal for students in Grades 10-12.

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