The length intercepted by the straight line `y =mx + c ` between the coordinate axes is

Options

A. `(c)/(m) sqrt (1 + m ^(2))`
B. `(c)/(m)`
C. `sqrt ( c ^(2) + m ^(2))`
D. None of these

Answer

To find the length intercepted by the straight line \( y = mx + c \) between the coordinate axes, we can follow these steps:

### Step 1: Identify the intercepts
The line \( y = mx + c \) intersects the y-axis when \( x = 0 \). By substituting \( x = 0 \) into the equation:
\[ y = m(0) + c = c \]
Thus, the y-intercept is the point \( A(0, c) \).

Next, we find the x-intercept by setting \( y = 0 \):
\[ 0 = mx + c \implies mx = -c \implies x = -\frac{c}{m} \]
Thus, the x-intercept is the point \( B\left(-\frac{c}{m}, 0\right) \).

### Step 2: Determine the coordinates of the intercepts
We have:
- Point A (y-intercept): \( (0, c) \)
- Point B (x-intercept): \( \left(-\frac{c}{m}, 0\right) \)

### Step 3: Calculate the length of the line segment AB
To find the distance \( AB \), we can use the distance formula:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of points A and B:
\[ AB = \sqrt{\left(-\frac{c}{m} - 0\right)^2 + (0 - c)^2} \]
This simplifies to:
\[ AB = \sqrt{\left(-\frac{c}{m}\right)^2 + (-c)^2} \]
\[ = \sqrt{\frac{c^2}{m^2} + c^2} \]
Factoring out \( c^2 \):
\[ = \sqrt{c^2\left(\frac{1}{m^2} + 1\right)} = c \sqrt{\frac{1}{m^2} + 1} \]

### Step 4: Simplify the expression
We can rewrite \( \frac{1}{m^2} + 1 \) as:
\[ \frac{1 + m^2}{m^2} \]
Thus, we have:
\[ AB = c \sqrt{\frac{1 + m^2}{m^2}} = c \cdot \frac{\sqrt{1 + m^2}}{m} \]
So, the final expression for the length intercepted by the line \( y = mx + c \) between the coordinate axes is:
\[ AB = \frac{c \sqrt{1 + m^2}}{m} \]

### Final Answer
The length intercepted by the straight line \( y = mx + c \) between the coordinate axes is:
\[ \frac{c \sqrt{1 + m^2}}{m} \]

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