find the ratio in which the line 2x + 3y - 5 = 0 divideds the line segment joining the - Math - Coordinate Geometry

Question

find the ratio in which the line 2x + 3y - 5 = 0 divideds the
line segment joining the points (8,-9) and (2,1) also find the
coordinates of the point of division

Shridhar Kaushik · Accepted Answer

Let the co-ordinates be P (x, y) Let the ratio be m : n in which the line 2x + 3y â€“ 5 = 0 divides the line segment joining the points (8, â€“9) and (2, 1) Since this point P lies on the line joining the points (8, â€“9) and (2, 1) and divide the line segment in the ratio m : n

So, the co-ordinates of point P are $x = \frac{2 m + 8 n}{m + n} and y = \frac{m - 9 n}{m + n} [By section formula]$ Since this point P also lies on the given line 2x + 3y â€“ 5 = 0, so it must satisfy the equation of this line so, $2 (\frac{2 m + 8 n}{m + n}) + 3 (\frac{m - 9 n}{m + n}) - 5 = 0 \Rightarrow 4 m + 16 n + 3 m - 27 n - 5 m - 5 n = 0 \Rightarrow 2 m - 16 n = 0 \Rightarrow 2 m = 16 n \Rightarrow \frac{m}{n} = \frac{8}{1}$ Hence m : n = 8: 1 and so the co-ordinates of point of division are $(\frac{2 \times 8 + 8 \times 1}{8 + 1}, \frac{8 - 9 \times 1}{8 + 1}) i e , (\frac{24}{9}, - \frac{1}{9})$

find the ratio in which the line 2x + 3y - 5 = 0 divideds the line segment joining the points (8,-9) and (2,1) . also find the coordinates of the point of division.