Find the distance of the point P from the lines AB in the following cases :
P(0, 0), AB is `h(x+h)+k(y+k)=0`

Answer

To find the distance of the point \( P(0, 0) \) from the line defined by the equation \( h(x + h) + k(y + k) = 0 \), we will follow these steps:

### Step 1: Rewrite the line equation
The given line equation is:
\[ h(x + h) + k(y + k) = 0 \]
Expanding this, we have:
\[ hx + h^2 + ky + k^2 = 0 \]
Rearranging gives:
\[ hx + ky + (h^2 + k^2) = 0 \]

### Step 2: Identify coefficients
From the equation \( hx + ky + (h^2 + k^2) = 0 \), we can identify:
- \( a = h \)
- \( b = k \)
- \( c = h^2 + k^2 \)

### Step 3: Use the distance formula
The distance \( d \) from a point \( (x_1, y_1) \) to the line \( ax + by + c = 0 \) is given by the formula:
\[ d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}} \]
For our point \( P(0, 0) \), we have \( x_1 = 0 \) and \( y_1 = 0 \). Substituting these values into the formula gives:
\[ d = \frac{|h(0) + k(0) + (h^2 + k^2)|}{\sqrt{h^2 + k^2}} = \frac{|h^2 + k^2|}{\sqrt{h^2 + k^2}} \]

### Step 4: Simplify the distance
Since \( h^2 + k^2 \) is always non-negative, we can drop the absolute value:
\[ d = \frac{h^2 + k^2}{\sqrt{h^2 + k^2}} \]
This can be further simplified:
\[ d = \sqrt{h^2 + k^2} \]

### Final Answer
Thus, the distance of the point \( P(0, 0) \) from the line \( AB \) is:
\[ \sqrt{h^2 + k^2} \]