The length of line segment is 3 which is perpendicular on line `4x+3y+C=0` from the origin. Then value of c will be

Options

A. 0
B. 7
C. 10
D. 15

Answer

To find the value of \( c \) in the line equation \( 4x + 3y + c = 0 \) given that the perpendicular distance from the origin to this line is 3, we can follow these steps:

### Step 1: Identify the coefficients
The line equation is in the form \( ax + by + c = 0 \). Here, we identify:
- \( a = 4 \)
- \( b = 3 \)
- \( c = c \) (unknown)

### Step 2: Use the formula for perpendicular distance
The formula for the perpendicular distance \( d \) from a point \( (x_1, y_1) \) to the line \( ax + by + c = 0 \) is given by:

\[ d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}} \]

### Step 3: Substitute the values
Since we are finding the distance from the origin \( (0, 0) \), we substitute \( x_1 = 0 \) and \( y_1 = 0 \):

\[ d = \frac{|4(0) + 3(0) + c|}{\sqrt{4^2 + 3^2}} = \frac{|c|}{\sqrt{16 + 9}} = \frac{|c|}{\sqrt{25}} = \frac{|c|}{5} \]

### Step 4: Set the distance equal to 3
According to the problem, the perpendicular distance \( d \) is 3. Therefore, we set up the equation:

\[ \frac{|c|}{5} = 3 \]

### Step 5: Solve for \( |c| \)
To solve for \( |c| \), we multiply both sides by 5:

\[ |c| = 3 \times 5 = 15 \]

### Step 6: Determine the value of \( c \)
Since \( |c| = 15 \), \( c \) can be either \( 15 \) or \( -15 \). However, the problem does not specify a sign for \( c \), so we can conclude:

\[ c = 15 \quad \text{(assuming we take the positive value)} \]

### Final Answer
Thus, the value of \( c \) is \( 15 \).

---