With oblique coordinates find the tangent of the angle between the straight lines
y = mx + c
my + x = d
To find the angle between the straight lines given by the equations y = mx + c and my + x = d, we can use the concept of oblique coordinates.
1. Solve the given equations simultaneously to find the point of intersection (x₀, y₀) where the two lines intersect. To do this, we can eliminate y by multiplying the first equation by m:
m(y) = m(mx + c)
my = m²x + mc (equation 1)
Substitute this value for my in the second equation:
m²x + mc + x = d
m²x + x = d - mc
(m² + 1)x = d - mc
x = (d - mc)/(m² + 1)
Substitute this value of x into the first equation to find y:
y = mx + c
y = m[(d - mc)/(m² + 1)] + c
y = (md - mc²)/(m² + 1)
So, the point of intersection is ((d - mc)/(m² + 1), (md - mc²)/(m² + 1)).
2. Calculate the slopes of the two lines. The slope of the first line, m₁, is simply the coefficient of x in the equation y = mx + c. The slope of the second line, m₂, is the negative reciprocal of the coefficient of y:
m₁ = m
m₂ = -1/m
3. Use the formula for the tangent of the angle between two lines:
tan(θ) = |(m₁ - m₂) / (1 + m₁ * m₂)|
Substituting the values of m₁ and m₂, we have:
tan(θ) = |(m - (-1/m)) / (1 + m * (-1/m))|
= |(m + 1/m) / (1 - m²)|
Simplify the expression inside the absolute value by multiplying numerator and denominator by m:
tan(θ) = |((m² + 1)/m) / (m² - 1)|
The tangent of the angle between the two lines is given by this expression.
Note: If the value of m is zero or undefined (vertical line), these formulas will not work as expected, as they involve division by m. In such cases, special handling is required.