Explanation:

To find the ratio in which the origin divides the line passing through it and perpendicular to the given lines, we can use the concept of slopes.

The equation of a line passing through the origin can be written as y = mx, where m is the slope of the line.

Step 1: Finding the slope of the given lines:
The given lines are:
1) 2x + y + 6 = 0
2) 4x + 2y - 9 = 0

To find the slope of a line in the form Ax + By + C = 0, we can rearrange the equation to the form y = mx + b, where m is the slope.

For the first line:
2x + y + 6 = 0
y = -2x - 6

The slope (m1) of the first line is -2.

For the second line:
4x + 2y - 9 = 0
2y = -4x + 9
y = -2x + 9/2

The slope (m2) of the second line is -2.

Step 2: Finding the slope of the line perpendicular to both lines:
The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line.

The slope of the line perpendicular to both lines is the negative reciprocal of the slopes of the given lines.

For the first line, the slope of the perpendicular line is 1/2.
For the second line, the slope of the perpendicular line is 1/2.

Step 3: Writing the equation of the perpendicular line:
We know that the equation of a line passing through the origin is y = mx.

The equation of the line perpendicular to both given lines and passing through the origin is y = (1/2)x.

Step 4: Finding the ratio in which the origin divides the line:
To find the ratio in which the origin divides the line, we need to find the distances from the origin to the points where the line intersects the x and y axes.

When the line intersects the x-axis, y = 0. Substituting y = 0 in the equation of the line, we get x = 0.

When the line intersects the y-axis, x = 0. Substituting x = 0 in the equation of the line, we get y = 0.

Therefore, the origin divides the line in the ratio of 0:0, which means it divides the line at the origin itself.

Conclusion:
The origin divides the line passing through it and perpendicular to the given lines at the ratio of 0:0.

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