Introduction

When dealing with geometric and trigonometric problems, understanding how to find the equation of a line that bisects the angle between the axes in specific quadrants can be quite useful. This article explores the process of determining the equation of a line that passes through the origin and bisects the angle between the positive x-axis in quadrant 1 and the negative y-axis in quadrant 3. We'll provide a clear, step-by-step solution to understand the underlying mathematics.

Identifying the Angles

To begin, let's establish the coordinate axes and the angles involved:

The positive x-axis corresponds to an angle of (theta 0^circ) or (theta 0) radians. The negative y-axis corresponds to an angle of (theta 270^circ) or (theta frac{3pi}{2}) radians.

Calculating the Angle Between the Axes

The angle between these two axes is:

[theta 270^circ - 0^circ 270^circ]

To find the angle bisector, we need to divide this angle by two:

[theta_{text{bisector}} frac{270^circ}{2} 135^circ]

This corresponds to an angle of (theta frac{3pi}{4}) radians.

Determining the Slope of the Bisector Line

The slope (m) of a line can be determined using the tangent of the angle it makes with the x-axis. The slope is given by:

[m tan(theta)]

For (theta_{text{bisector}} 135^circ), we have:

[tan(135^circ) tanleft(180^circ - 45^circright) -tan(45^circ) -1]

Therefore, the slope of the line is (-1).

Equation of the Line

Since the line passes through the origin, the equation of the line can be expressed in point-slope form:

[y mx b]

Given that the line passes through the origin, (b 0). Therefore, the equation simplifies to:

[y -x]

This is the equation of the line that passes through the origin and bisects the angle between the positive x-axis in quadrant 1 and the negative y-axis in quadrant 3.

Verification

To ensure the solution is correct, we can verify the following:

The line passes through the origin, as ((0, 0)) satisfies (y -x) when (x 0). The slope of the line is (-1), which means the angle the line makes with the x-axis is (135^circ), or (frac{3pi}{4}) radians. The angle between the x and y axes is (90^circ), or (frac{pi}{2}) radians. The angle the line makes with the y-axis is: [text{Angle with y-axis} frac{pi}{2} - frac{3pi}{4} frac{pi}{4}]

Thus, the line bisects the angle between the positive x-axis and positive y-axis, as well as the angle between the positive y-axis and negative x-axis.

Conclusion

Therefore, the equation of the line that passes through the origin and bisects the angle between the axes in quadrant 1 and 3 is:

[boxed{y -x}]