Find the length of perpendicular from the origin to each of the following (i)7x+24y=50 (ii)4x+3y=9 (iii)x=4

Explanation

To find the length of the perpendicular from the origin to each of the given lines, we will use the formula for the distance from a point to a line. The formula for the perpendicular distance \( D \) from a point \( (x_1, y_1) \) to the line \( Ax + By + C = 0 \) is given by:

\[ D = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]

Where \( A \), \( B \), and \( C \) are the coefficients from the line equation.

### (i) For the line \( 7x + 24y = 50 \)

1. **Rewrite the line equation in standard form**:
\[ 7x + 24y - 50 = 0 \]
Here, \( A = 7 \), \( B = 24 \), and \( C = -50 \).

2. **Substitute the origin coordinates \( (0, 0) \) into the distance formula**:
\[ D = \frac{|7(0) + 24(0) - 50|}{\sqrt{7^2 + 24^2}} = \frac{|-50|}{\sqrt{49 + 576}} = \frac{50}{\sqrt{625}} \]

3. **Calculate the denominator**:
\[ \sqrt{625} = 25 \]

4. **Final calculation for distance**:
\[ D = \frac{50}{25} = 2 \]

### (ii) For the line \( 4x + 3y = 9 \)

1. **Rewrite the line equation in standard form**:
\[ 4x + 3y - 9 = 0 \]
Here, \( A = 4 \), \( B = 3 \), and \( C = -9 \).

2. **Substitute the origin coordinates \( (0, 0) \)**:
\[ D = \frac{|4(0) + 3(0) - 9|}{\sqrt{4^2 + 3^2}} = \frac{|-9|}{\sqrt{16 + 9}} = \frac{9}{\sqrt{25}} \]

3. **Calculate the denominator**:
\[ \sqrt{25} = 5 \]

4. **Final calculation for distance**:
\[ D = \frac{9}{5} \]

### (iii) For the line \( x = 4 \)

1. **Identify the line**:
The line \( x = 4 \) is vertical. The distance from the origin \( (0, 0) \) to this line is simply the horizontal distance.

2. **Calculate the distance**:
The distance is:
\[ D = |4 - 0| = 4 \]

### Summary of Results:
- The length of the perpendicular from the origin to the line \( 7x + 24y = 50 \) is **2**.
- The length of the perpendicular from the origin to the line \( 4x + 3y = 9 \) is **\( \frac{9}{5} \)**.
- The length of the perpendicular from the origin to the line \( x = 4 \) is **4**.