In the given figure, the line segment AB meets x-axis at point A and y-axis at point B.

The point P(-5, 3) on AB divides it in the ratio 3 : 2. Find :

(i) co-ordinates of point A

(ii) co-ordinates of point B

(iii) equation of line through point P and perpendicular to AB.

(iv) the point of intersection of the line (obtained in part (iii) above) and x-axis.

Let co-ordinates of A be (a, 0) and B be (0, b).

Given, P(-5, 3) on AB divides it in the ratio 3 : 2.

By section formula,

(x, y) = $\Big(\dfrac{m$1x2 + m2x1}{m1 + m2}, \dfrac{m1y2 + m2y1}{m1 + m2}\Big)

Substituting values we get :

$\Rightarrow (-5, 3) = \Big(\dfrac{3 \times 0 + 2 \times a}{3 + 2}, \dfrac{3 \times b + 2 \times 0}{3 + 2}\Big) \\[1em] \Rightarrow (-5, 3) = \Big(\dfrac{2a}{5}, \dfrac{3b}{5}\Big) \\[1em] \Rightarrow -5 = \dfrac{2a}{5} \text{ and } 3 = \dfrac{3b}{5} \\[1em] \Rightarrow a = \dfrac{-5 \times 5}{2} \text{ and } b = \dfrac{3 \times 5}{3} \\[1em] \Rightarrow a = -\dfrac{25}{2} \text{ and } b = 5.$

A = (a, 0) = $\Big(-\dfrac{25}{2}, 0\Big)$

B = (0, b) = (0, 5).

(i) Hence, co-ordinates of A = $\Big(-\dfrac{25}{2}, 0\Big)$.

(ii) Hence, co-ordinates of B = (0, 5).

(iii) By formula,

Slope = $\dfrac{y$2 - y1}{x2 - x1}

Slope of AB

= $\dfrac{5 - 0}{0 - \Big(-\dfrac{25}{2}\Big)} = \dfrac{5}{\dfrac{25}{2}} = \dfrac{5 \times 2}{25} = \dfrac{2}{5}$.

We know that,

Product of slope of perpendicular lines = -1.

Let slope of line perpendicular to AB be a.

⇒ $a \times \dfrac{2}{5}$ = -1

⇒ a = $-\dfrac{5}{2}$.

By point-slope form,

Equation of line : y - y₁ = m(x - x₁)

Substituting values we get :

⇒ y - 3 = $-\dfrac{5}{2}$ [x - (-5)]

⇒ 2(y - 3) = -5(x + 5)

⇒ 2y - 6 = -5x - 25

⇒ 2y + 5x - 6 + 25 = 0

⇒ 2y + 5x + 19 = 0.

Hence, equation of line through point P and perpendicular to AB is 2y + 5x + 19 = 0.

(iv) Any point on x-axis is given by (x₁, 0).

Substituting values in equation 2y + 5x + 19 = 0, we get :

⇒ 2(0) + 5x₁ + 19 = 0

⇒ 5x₁ + 19 = 0

⇒ 5x₁ = -19

⇒ x₁ = $-\dfrac{19}{5}$

⇒ (x₁, 0) = $\Big(-\dfrac{19}{5}, 0\Big)$.

Hence, point of intersection = $\Big(-\dfrac{19}{5}, 0\Big)$.