What is the distnace of the line `3x - 4y + 15 =0` from origin ? What is its distance from point `(-5, -1).`

Answer

To find the distance of the line \(3x - 4y + 15 = 0\) from the origin and from the point \((-5, -1)\), we can follow these steps:

### Step 1: Identify the coefficients
The equation of the line is given in the form \(Ax + By + C = 0\). Here, we can identify:
- \(A = 3\)
- \(B = -4\)
- \(C = 15\)

### Step 2: Calculate the distance from the origin
The formula to find the distance \(d\) from the line \(Ax + By + C = 0\) to the origin \((0, 0)\) is given by:
\[ d = \frac{|C|}{\sqrt{A^2 + B^2}} \]
Substituting the values we identified:
\[ d = \frac{|15|}{\sqrt{3^2 + (-4)^2}} = \frac{15}{\sqrt{9 + 16}} = \frac{15}{\sqrt{25}} = \frac{15}{5} = 3 \]
Thus, the distance from the origin is **3 units**.

### Step 3: Calculate the distance from the point \((-5, -1)\)
To find the distance from a point \((x_1, y_1)\) to the line \(Ax + By + C = 0\), we use the formula:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
Substituting \(x_1 = -5\) and \(y_1 = -1\):
\[ d = \frac{|3(-5) + (-4)(-1) + 15|}{\sqrt{3^2 + (-4)^2}} = \frac{|-15 + 4 + 15|}{\sqrt{9 + 16}} = \frac{|4|}{\sqrt{25}} = \frac{4}{5} \]
Thus, the distance from the point \((-5, -1)\) is **\(\frac{4}{5}\) units**.

### Summary of Results
- Distance from the origin: **3 units**
- Distance from the point \((-5, -1)\): **\(\frac{4}{5}\) units**

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