Straight Lines

We need to find the equation of a straight line that passes through the point and divides the segment between two points (not specified in the query) in the ratio 1:2. Since the two points are not given, we interpret the problem as finding the line passing through such that it divides the segment between two points and (to be determined) in the ratio 1:2 at .

1. Interpret the Ratio Condition:
The point divides the segment between two points and in the ratio 1:2. Without and , let’s assume the problem might mean the line’s intercepts on the axes, or we need to find a line where divides some segment. A common interpretation is that lies on the line, and the ratio might refer to a segment defined by intercepts or another point. Let’s proceed by assuming we need the line equation, and the ratio might apply to a segment defined later. First, let’s find a general line through .

2. General Equation of the Line:
A line passing through the point can be written in the point-slope form:


Using :




This is the equation of the line with slope , which we need to determine using the ratio condition.

3. Find the Intercepts of the Line:
From the line equation :
- X-intercept: Set :




So, the x-intercept is . If , the line is , which has no x-intercept, so assume .
- Y-intercept: Set :


So, the y-intercept is .

4. Apply the Ratio Condition:
Assume divides the segment between the x-intercept (point ) and the y-intercept (point ) in the ratio 1:2. Using the section formula, if a point divides the segment joining and in the ratio , its coordinates are:


Here, , , and the ratio is 1:2, so . The coordinates of the point dividing should match :

- X-coordinate:











- Y-coordinate:







Both coordinates give the same , confirming consistency.

5. Equation of the Line:
Substitute into the line equation:




Multiply through by 5 to clear the fraction:

6. Verify:
- The line passes through :

, which holds.
- Check intercepts:

X-intercept ( ): , , so point .
Y-intercept ( ): , , so point .
- Ratio check: divides to in 1:2:

X: , matches.
Y: , matches.

Final Answer:
The equation of the straight line is: