When the curve is given by y = f(x) then the slope of the tangent is so by the point–slope formula the equation of the tangent line at (X, Y) is where (x, y) are the coordinates of any point on the tangent line, and where the derivative is evaluated at . When the curve is given by y = f(x), the tangent line's equation can also be found by using polynomial division to divide by ; if the remainder is denoted by , then the equation of the tangent line is given by When the equation of the curve is given in the form f(x, y) = 0 then the value of the slope can be found by implicit differentiation, giving The equation of the tangent line at a point (X,Y) such that f(X,Y) = 0 is then This equation remains true if but (in this case the slope of the tangent is infinite). If the tangent line is not defined and the point (X,Y) is said to be singular. For algebraic curves, computations may be simplified somewhat by converting to homogeneous coordinates. Specifically, let the homogeneous equation of the curve be g(x, y, z) = 0 where g is a homogeneous function of degree n. Then, if (X, Y, Z) lies on the curve, Euler's theorem implies It follows that the homogeneous equation of the tangent line is The equation of the tangent line in Cartesian coordinates can be found by setting z=1 in this equation. To apply this to algebraic curves, write f(x, y) as where each ur is the sum of all terms of degree r. The homogeneous equation of the curve is then Applying the equation above and setting z=1 produces as the equation of the tangent line. The equation in this form is often simpler to use in practice since no further simplification is needed after it is applied. If the curve is given parametrically by then the slope of the tangent is giving the equation for the tangent line at as If the tangent line is not defined. However, it may occur that the tangent line exists and may be computed from an implicit equation of the curve. Normal line to a curve The line perpendicular to the tangent line to a curve at the point of tangency is called the normal line to the curve at that point. The slopes of perpendicular