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.