Question

the locus of a point which divides the join of a(-1,1) and a variable point p on the circle x square = y square =4 in the ratio 3:2 is ?? 

Answer 1

Let a point on circle be P (x1, y1).

The coordinates of point Q(x, y) which divides line between (-1,1) & P(x1, y1) in ratio 3 : 2 or 3/5 : 2/5 are :

x = (-1 * 2 + x1 * 3 )/5   So,  (5x +2)/3 = x1

y = (1 * 2 + y1 * 3) / 5    so,  y1 = (5y-2)/3

P(x1,y1) is on circle:

x1² + y1² = 2²

(5x + 2)² / 9 + (5y-2)² / 9 = 2²

(x + 2/5)² + ( y - 2/5)² = 36 / 25

The locus is a circle with center at (-2/5, 2/5) and radius = 6/5