A line passing through `(0,0)` and perpendicular to `2x+y+6=0,4x+2y-9=0` then the origin divids the line in the ratio of

Options

A. `1:2`
B. `2:1`
C. `4:3`
D. `3:4`

Answer

To solve the problem step by step, we will find the equation of the line that passes through the origin (0,0) and is perpendicular to the given lines. Then, we will find the ratio in which the origin divides this line.

### Step 1: Identify the given lines
The equations of the lines are:
1. \( 2x + y + 6 = 0 \)
2. \( 4x + 2y - 9 = 0 \)

### Step 2: Determine the slopes of the given lines
To find the slope of a line in the form \( Ax + By + C = 0 \), we can use the formula:
\[ \text{slope} = -\frac{A}{B} \]

For the first line \( 2x + y + 6 = 0 \):
- Coefficient of \( x \) (A) = 2
- Coefficient of \( y \) (B) = 1
\[ \text{slope of line 1} = -\frac{2}{1} = -2 \]

For the second line \( 4x + 2y - 9 = 0 \):
- Coefficient of \( x \) (A) = 4
- Coefficient of \( y \) (B) = 2
\[ \text{slope of line 2} = -\frac{4}{2} = -2 \]

Both lines have the same slope, which means they are parallel.

### Step 3: Find the slope of the perpendicular line
If two lines are perpendicular, the product of their slopes is -1. Let \( m_1 = -2 \) (slope of the given lines) and \( m_2 \) be the slope of the line we need to find.
\[ m_1 \cdot m_2 = -1 \implies -2 \cdot m_2 = -1 \implies m_2 = \frac{1}{2} \]

### Step 4: Write the equation of the perpendicular line
Using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Since the line passes through the origin (0,0):
\[ y - 0 = \frac{1}{2}(x - 0) \implies y = \frac{1}{2}x \]
Rearranging gives:
\[ x - 2y = 0 \]

### Step 5: Find the intersection points of the lines
To find the points where the line \( x - 2y = 0 \) intersects the given lines, we will solve for the intersection points.

**Intersection with \( 2x + y + 6 = 0 \)**:
Substituting \( y = \frac{1}{2}x \) into \( 2x + y + 6 = 0 \):
\[ 2x + \frac{1}{2}x + 6 = 0 \implies \frac{5}{2}x + 6 = 0 \implies \frac{5}{2}x = -6 \implies x = -\frac{12}{5} \]
Then, substituting back to find \( y \):
\[ y = \frac{1}{2}(-\frac{12}{5}) = -\frac{6}{5} \]
So, the intersection point is \( \left(-\frac{12}{5}, -\frac{6}{5}\right) \).

**Intersection with \( 4x + 2y - 9 = 0 \)**:
Substituting \( y = \frac{1}{2}x \) into \( 4x + 2y - 9 = 0 \):
\[ 4x + 2(\frac{1}{2}x) - 9 = 0 \implies 4x + x - 9 = 0 \implies 5x = 9 \implies x = \frac{9}{5} \]
Then, substituting back to find \( y \):
\[ y = \frac{1}{2}(\frac{9}{5}) = \frac{9}{10} \]
So, the intersection point is \( \left(\frac{9}{5}, \frac{9}{10}\right) \).

### Step 6: Use the section formula
Let the origin divide the line segment joining the two intersection points in the ratio \( \lambda:1 \). The coordinates of the points are:
1. \( A\left(-\frac{12}{5}, -\frac{6}{5}\right) \)
2. \( B\left(\frac{9}{5}, \frac{9}{10}\right) \)

Using the section formula:
\[ \left( \frac{\lambda x_2 + x_1}{\lambda + 1}, \frac{\lambda y_2 + y_1}{\lambda + 1} \right) = (0, 0) \]

For the x-coordinates:
\[ \frac{\lambda \cdot \frac{9}{5} - \frac{12}{5}}{\lambda + 1} = 0 \implies \lambda \cdot \frac{9}{5} - \frac{12}{5} = 0 \implies \lambda \cdot 9 = 12 \implies \lambda = \frac{12}{9} = \frac{4}{3} \]

Thus, the ratio in which the origin divides the line is \( 4:3 \).

### Final Answer
The origin divides the line in the ratio \( 4:3 \).