generally, in the plane cartesian coordinate system, if the straight line l passes through the points a(x1,y1) and b(x2,y2), where x1≠x2, then ab=(x2-x1,y2-y1) is a direction vector, so the slope of straight line l k=(y2-y1)/(x2-x1), and then k=tanα (0≤α<π), the inclination angle α of straight line l can be calculated. note tanα=k, the equation y-y0=k(x-x0) is called the point-slope equation of the straight line, where (x0, y0) is a point on the straight line.

when α is π/2 (90 degrees, the straight line is perpendicular to the x-axis), tanα is meaningless, and there is no point-slope equation.

point-slope equations are commonly used in derivatives. they are used to find tangent equations using a known point on the tangent and the derivative of the curve equation (the slope of the tangent line at a certain point on the equation). it is suitable for questions where you know the coordinates of a point and the slope of a straight line and find the equation of the straight line.