Find equation of the perpendicular to segment joining the points A(0,4) and B(-5,9) and passing through the point P. Point P divides segment AB in the ratio 2:3.

Using section formula, the coordinates of point that divides line joining A = $$(x_1 , y_1)$$ and B = $$(x_2 , y_2)$$ in the ratio a : b

= $$(\frac{a x_2 + b x_1}{a + b} , \frac{a y_2 + b y_1}{a + b})$$

Coordinates of A(0,4) and B(-5,9). Let coordinates of P = (x,y) which divides AB in ratio = 2 : 3

=> $$x = \frac{(2 \times -5) + (3 \times 0)}{2 + 3}$$

=> $$5x = -10$$

=> $$x = \frac{-10}{5} = -2$$

Similarly, $$y = \frac{(2 \times 9) + (3 \times 4)}{2 + 3}$$

=> $$5y = 18 + 12 = 30$$

=> $$y = \frac{30}{5} = 6$$

=> Point P = (-2,6)

Slope of AB = $$\frac{9 - 4}{-5 - 0} = \frac{5}{-5} = -1$$

Let slope of line perpendicular to AB = $$m$$

Also, product of slopes of two perpendicular lines is -1

=> $$m \times -1 = -1$$

=> $$m = 1$$

Equation of lines having slope $$m$$ and passing through point P(-2,6) is

=> $$(y - 6) = 1(x + 2)$$

=> $$y - 6 = x + 2$$

=> $$x - y = -8$$

=> Ans - (B)