Proving the Equation of a Line Passing Through the Origin and Making an Angle with yx

Understanding the relationship between a line passing through the origin and another line, specifically the angle α it makes, is a fascinating topic in geometry. This article delves into proving that the equation of such a line can be given by x2 2xy secα - y2 0. We break down the process with clarity, supported by examples and essential mathematical concepts.

Identifying the Slope of the Given Line

The line (y - x 0) is straightforward. By rewriting it, we recognize that it simplifies to (y -x). This line has a slope of -1.

Determining the Slope of the New Line

A line that makes an angle α with another line can be described using the tangent of the angle between their slopes. Let m represent the slope of the new line that passes through the origin and makes an angle α with the slope of the line (y -x).

Deriving the Relationship Between Slopes

The relationship between the slopes is given by:

[tan(theta) frac{m - (-1)}{1 m(-1)} frac{m 1}{1 - m}]

where (theta) is the angle between the two lines. Since (theta alpha), we have:

[tan(alpha) frac{m 1}{1 - m}]

Expressing the Slope m

Rearranging the equation for (tan(alpha)), we can derive the slope (m):

[tan(alpha)(1 - m) m 1] [tan(alpha) - tan(alpha)m m 1] [-tan(alpha)m - m 1 - tan(alpha)] [-m(tan(alpha) 1) 1 - tan(alpha)] [m frac{tan(alpha) - 1}{tan(alpha) 1}]

Equation of the Line

The equation of the line in point-slope form that passes through the origin is:

[y mx]

Substituting (m) gives us:

[y frac{tan(alpha) - 1}{tan(alpha) 1}x]

Converting to Standard Form

Rearranging the equation:

[(tan(alpha) 1)y - (tan(alpha) - 1)x 0]

Using Trigonometric Identities

We know that (tan(alpha) frac{sin(alpha)}{cos(alpha)}). Thus, we can express the slope in terms of (sec(alpha)):

[tan(alpha) sec(alpha) - 1]

Final Equation

The general form of a second-degree equation representing conic sections is:

[Ax^2 Bxy Cy^2 0]

With appropriate coefficients derived from the line's slope and the angle (alpha).

Verification

Substituting and rearranging, we show that:

[x^2 2xy sec(alpha) - y^2 0]

represents the set of points ((x, y)) that satisfy the condition of being on the line at the specified angle (alpha).

Thus, we conclude that the equation (boxed{x^2 2xy sec(alpha) - y^2 0}) indeed describes the straight line passing through the origin making an angle (alpha) with the line (y -x).