Thank you for sharing the image. I’ll analyze it and extract the mathematical content to provide solutions. Let me begin interpreting the equations and problems. One moment, please.The image contains three math problems written in Persian. I will solve each one step by step:

Problem 1

If $f(x) = ax^r + b$ passes through the point $A(1, -2)$ and is symmetric around the origin, find $\frac{b}{a}$.

• Symmetry around the origin means $f(-x) = -f(x)$.

• Substituting this property into $f(x)$:
  $a(-x)^r + b = -(ax^r + b)$
  Simplifying:
  $(-1)^r ax^r + b = -ax^r - b$
  Therefore:
  $(-1)^r a = -a$ and $b = -b$.
  From $b = -b$, we find $b = 0$.

• Substituting $A(1, -2)$ into $f(x)$:
  $-2 = a(1)^r + 0$
  $a = -2$.

Thus, $\frac{b}{a} = \frac{0}{-2} = 0$.

Answer: Option 4.

Problem 2

Given rectangle $ABCD$ with width 8 and height 6, and $AC$ as a diagonal, $B$ is at an angle bisector, and $E$ lies on $AC$. Find the length of $AE$.

• The rectangle dimensions are given as 8 (width) and 6 (height).
• Diagonal $AC = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = 10$.
• $B$ lies on the angle bisector, and $E$ divides $AC$ into segments proportional to the rectangle's sides.

Let the ratio of division of $AC$ be proportional to $8:6 = 4:3$.

• Therefore:
  $AE = \frac{4}{4+3} \cdot AC = \frac{4}{7} \cdot 10 = \frac{40}{7}$.

Answer: Option 3.

Problem 3

If $f(x) = \frac{Fx + d}{x - b}$ and $g(x) = \frac{2}{x - b}$, find the difference of coefficients that leads to equal functions.

• Comparing $f(x)$ and $g(x)$:
  $\frac{Fx + d}{x - b} = \frac{2}{x - b}$.
  Equating the numerators since the denominators are the same:
  $Fx + d = 2$.
  • Coefficient of $x$ is $F$, and the constant term is $d$.

From $Fx + d = 2$:
$F = 0$ and $d = 2$.
The difference between coefficients $F - d = 0 - 2 = -2$.

Answer: Option 4.

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Related Questions

1. What does symmetry around the origin imply in a function graphically?
2. How do you derive the formula for diagonals in a rectangle?
3. Why is the division of the diagonal $AC$ in the ratio of rectangle sides?
4. How do you compare two rational functions to find coefficients?
5. What are the key steps in solving problems involving ratios in geometry?

Tip: When working with ratios, always reduce to the simplest form before calculating.