Mathematics

A line is tangent to a curve when it touches it at exactly one point—meaning the system of equations has exactly one solution. Substitute the line equation into the hyperbola and you'll get a quadratic in $x$; for tangency, its discriminant must equal zero. This condition pins down the value of $\lambda$ that makes the line kiss the hyperbola rather than cross it.

A line is tangent to a curve when it touches it at exactly one point—meaning the system of equations has exactly one solution. Substitute the line equation into the hyperbola and you'll get a quadratic in $x$; for tangency, its discriminant must equal zero. This condition pins down the value of $\lambda$ that makes the line kiss the hyperbola rather than cross it.