Reduce the following equations to the normal form and find the values of p and `alpha`.
`3x+4y+10=0` (use tables).

Answer

To reduce the equation \(3x + 4y + 10 = 0\) to its normal form and find the values of \(p\) and \(\alpha\), we will follow these steps:

### Step 1: Rewrite the equation
Start with the given equation:
\[ 3x + 4y + 10 = 0 \]
Rearranging gives:
\[ 3x + 4y = -10 \]

### Step 2: Identify coefficients
From the equation \(3x + 4y = -10\), we identify the coefficients:
- Coefficient of \(x\) (denote as \(A\)) = 3
- Coefficient of \(y\) (denote as \(B\)) = 4

### Step 3: Calculate the denominator
We need to find the value of \(\sqrt{A^2 + B^2}\):
\[ \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 4: Divide the equation by the denominator
Now, divide the entire equation by 5:
\[ \frac{3}{5}x + \frac{4}{5}y = -\frac{10}{5} \]
This simplifies to:
\[ \frac{3}{5}x + \frac{4}{5}y = -2 \]

### Step 5: Rewrite in normal form
We can express this in the form:
\[ x \cos \alpha + y \sin \alpha = p \]
where:
- \(\cos \alpha = \frac{3}{5}\)
- \(\sin \alpha = \frac{4}{5}\)
- \(p = -2\)

### Step 6: Find \(p\)
From the equation, we see that:
\[ p = -2 \]

### Step 7: Find \(\alpha\)
To find \(\alpha\), we can use the relationship:
\[ \tan \alpha = \frac{\sin \alpha}{\cos \alpha} = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3} \]
Thus, we can find \(\alpha\) as:
\[ \alpha = \tan^{-1}\left(\frac{4}{3}\right) \]

### Final Result
The values are:
- \(p = -2\)
- \(\alpha = \tan^{-1}\left(\frac{4}{3}\right)\)