A point P(h, k) lies on the straight line `x + y +1 = 0` and is at a distance 5 from the origin. If k is negative, then h is equal to

Options

A. `-3`
B. 3
C. `-4`
D. 4

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step-by-Step Solution:

1. **Identify the equations**:
We know that the point \( P(h, k) \) lies on the line given by the equation:
\[ x + y + 1 = 0 \]
This can be rewritten as:
\[ k = -h - 1 \]

2. **Distance from the origin**:
The distance \( d \) from the origin \( (0, 0) \) to the point \( P(h, k) \) is given as 5. The distance formula is:
\[ d = \sqrt{(h - 0)^2 + (k - 0)^2} = 5 \]
Squaring both sides, we get:
\[ h^2 + k^2 = 25 \]

3. **Substituting for \( k \)**:
We substitute \( k \) from the line equation into the distance equation:
\[ h^2 + (-h - 1)^2 = 25 \]
Expanding the equation:
\[ h^2 + (h^2 + 2h + 1) = 25 \]
Simplifying this gives:
\[ 2h^2 + 2h + 1 = 25 \]

4. **Rearranging the equation**:
Rearranging the equation leads to:
\[ 2h^2 + 2h + 1 - 25 = 0 \]
This simplifies to:
\[ 2h^2 + 2h - 24 = 0 \]
Dividing the entire equation by 2:
\[ h^2 + h - 12 = 0 \]

5. **Factoring the quadratic**:
We can factor the quadratic equation:
\[ (h + 4)(h - 3) = 0 \]
This gives us two possible solutions for \( h \):
\[ h + 4 = 0 \quad \text{or} \quad h - 3 = 0 \]
Thus, \( h = -4 \) or \( h = 3 \).

6. **Finding the value of \( k \)**:
We need to check the value of \( k \) for both values of \( h \):
- If \( h = -4 \):
\[ k = -(-4) - 1 = 3 \]
- If \( h = 3 \):
\[ k = -(3) - 1 = -4 \]

7. **Considering the condition \( k < 0 \)**:
Since we are given that \( k \) is negative, we take the solution where \( h = 3 \) and \( k = -4 \).

### Final Answer:
Thus, the value of \( h \) is:
\[ \boxed{3} \]