Concept of a Locus When a particle moves on a plane under certain restrictions, it will move along a certain path. The path traced out by the moving particle is called locus.
e.g. when a particle is thrown obliquely upwards under gravitation, its locus will be a parabola.
e.g. an object is connected to a fixed point on a smooth floor with an inextensible string, when it is projected along a direction perpendicular to the string, it locus will be a circle.
Definition : • If a point which moves under certain conditions • describes a path, and • all points satisfying the conditions lie on the path, • every point on the path satisfies the conditions, • Then the path is called the locus of the point.
Example Find the equation of the locus of a point which is equidistant from the two points A(-1,0) and B(3,1). y P(x,y) B(3,1) O A(-1,0) x
y P(x,y) B(3,1) O A(-1,0) x 8x + 2y – 9 = 0 P’(x,y)
y P(x,y) 3 A(-1,2) O x Example Find the equation of the locus of a point P with distance 3 units from A(-1,2).
y P(x,y) 3 A(-1,2) O 3 x P’(x,y)
Exercise 10.1 P.113
Parametric Equations Let us consider the two equations : x = t2 + 2t, y = t - 1 x = f(t) y = f(t) parametric equations parameter
y x = t2 + 2t, y = t - 1 x
y x = y2 + 4y + 3 x
To find the locus of a certain path from parametric equations locus of a path parametric equations eliminate the parameter
Example For any real value of θ, P is the point (h + r cosθ, k + r sinθ), where h, k and r are constants. As θ varies, find the equation of the locus of P. Let P be (x,y), then
y P(h + rcosθ,k + rsinθ) (h, k) θ k O h x
Exercise 10.2 P.118
Exercise 10.3 P.124