### 1. Break Down the Problem
We have two lines represented in symmetric form. The goal is to find the relation between $\alpha$ and $\beta$ given that the angle between the two lines is $\frac{\pi}{4}$.

1. The first line is represented as:
$$
\frac{x - 5}{\alpha} = \frac{y + 2}{-5} = \frac{z + 24}{\beta}
$$
Which gives direction ratios: $\langle \alpha, -5, \beta \rangle$.

2. The second line is:
$$
\frac{x}{1} = \frac{y}{0} = \frac{z}{1}
$$
Which gives direction ratios: $\langle 1, 0, 1 \rangle$.

### 2. Relevant Concepts
To find the angle $ \theta $ between two lines, we use the formula:
$$
\cos(\theta) = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|}
$$
Where $\vec{a}$ and $\vec{b}$ are the direction ratios of the two lines. Given $ \theta = \frac{\pi}{4} $, we know that:
$$
\cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}
$$

### 3. Analysis and Detail

1. **Direction Ratios**:
- For Line 1: $\vec{a} = \langle \alpha, -5, \beta \rangle$
- For Line 2: $\vec{b} = \langle 1, 0, 1 \rangle$

2. **Dot Product**:
$$
\vec{a} \cdot \vec{b} = \alpha \cdot 1 + (-5) \cdot 0 + \beta \cdot 1 = \alpha + \beta
$$

3. **Magnitude of Vectors**:
- Magnitude of $\vec{a}$:
$$
|\vec{a}| = \sqrt{\alpha^2 + (-5)^2 + \beta^2} = \sqrt{\alpha^2 + 25 + \beta^2}
$$
- Magnitude of $\vec{b}$:
$$
|\vec{b}| = \sqrt{1^2 + 0^2 + 1^2} = \sqrt{2}
$$

4. **Substituting into the Cosine Formula**:
$$
\frac{\alpha + \beta}{\sqrt{\alpha^2 + 25 + \beta^2} \cdot \sqrt{2}} = \frac{1}{\sqrt{2}}
$$

### 4. Verify and Summarize

Multiply both sides by $\sqrt{2}$:
$$
\alpha + \beta = \sqrt{\alpha^2 + 25 + \beta^2}
$$

Now squaring both sides:
$$
(\alpha + \beta)^2 = \alpha^2 + 25 + \beta^2
$$

Expanding the left side:
$$
\alpha^2 + 2\alpha\beta + \beta^2 = \alpha^2 + 25 + \beta^2
$$

Canceling $\alpha^2$ and $\beta^2$ from both sides, we get:
$$
2\alpha\beta = 25 \quad \Rightarrow \quad \alpha \beta = \frac{25}{2}
$$

### Final Answer
The relation between $\alpha$ and $\beta$ is:
$$
\alpha \beta = \frac{25}{2}
$$

Question

### 1. Break Down the Problem
We have two lines represented in symmetric form. The goal is to find the relation between $\alpha$ and $\beta$ given that the angle between the two lines is $\frac{\pi}{4}$.

1. The first line is represented as:
   $$
   \frac{x - 5}{\alpha} = \frac{y + 2}{-5} = \frac{z + 24}{\beta}
   $$
   Which gives direction ratios: $\langle \alpha, -5, \beta \rangle$.

2. The second line is:
   $$
   \frac{x}{1} = \frac{y}{0} = \frac{z}{1}
   $$
   Which gives direction ratios: $\langle 1, 0, 1 \rangle$.

### 2. Relevant Concepts
To find the angle $ \theta $ between two lines, we use the formula:
$$
\cos(\theta) = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|}
$$
Where $\vec{a}$ and $\vec{b}$ are the direction ratios of the two lines. Given $ \theta = \frac{\pi}{4} $, we know that:
$$
\cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}
$$

### 3. Analysis and Detail

1. **Direction Ratios**:
   - For Line 1: $\vec{a} = \langle \alpha, -5, \beta \rangle$
   - For Line 2: $\vec{b} = \langle 1, 0, 1 \rangle$

2. **Dot Product**:
$$
\vec{a} \cdot \vec{b} = \alpha \cdot 1 + (-5) \cdot 0 + \beta \cdot 1 = \alpha + \beta
$$

3. **Magnitude of Vectors**:
   - Magnitude of $\vec{a}$:
$$
|\vec{a}| = \sqrt{\alpha^2 + (-5)^2 + \beta^2} = \sqrt{\alpha^2 + 25 + \beta^2}
$$
   - Magnitude of $\vec{b}$:
$$
|\vec{b}| = \sqrt{1^2 + 0^2 + 1^2} = \sqrt{2}
$$

4. **Substituting into the Cosine Formula**:
$$
\frac{\alpha + \beta}{\sqrt{\alpha^2 + 25 + \beta^2} \cdot \sqrt{2}} = \frac{1}{\sqrt{2}}
$$

### 4. Verify and Summarize

Multiply both sides by $\sqrt{2}$:
$$
\alpha + \beta = \sqrt{\alpha^2 + 25 + \beta^2}
$$

Now squaring both sides:
$$
(\alpha + \beta)^2 = \alpha^2 + 25 + \beta^2
$$

Expanding the left side:
$$
\alpha^2 + 2\alpha\beta + \beta^2 = \alpha^2 + 25 + \beta^2
$$

Canceling $\alpha^2$ and $\beta^2$ from both sides, we get:
$$
2\alpha\beta = 25 \quad \Rightarrow \quad \alpha \beta = \frac{25}{2}
$$

### Final Answer
The relation between $\alpha$ and $\beta$ is:
$$
\alpha \beta = \frac{25}{2}
$$

Knowee AI · Accepted Answer