The line segment joining A(2, 3) and B(6, -5) is intersected by x-axis at a point K. Write down the ordinate of the point K. Hence, find the ratio in which K divides AB. Also find the coordinates of point K.

Let the coordinates of K be (x, 0) as it intersects x-axis. Let point K divides the line segment joining the points
A(2, 3) and B(6, -5) in the ratio m₁ : m₂.

By section formula, y-coordinate = $\Big(\dfrac{m$1y2 + m2y1}{m1 + m2}\Big)

Putting value in above formula we get,

$\Rightarrow 0 = \dfrac{m$1 \times (-5) + m2 \times 3}{m1 + m2} \\[1em] \Rightarrow 0 = -5m1 + 3m2 \\[1em] \Rightarrow 5m1 = 3m2 \\[1em] \Rightarrow \dfrac{m1}{m2} = \dfrac{3}{5}

Now for x coordinate by section formula we get,

$x = \dfrac{m$1x2 + m2x1}{m1 + m2} \\[1em] = \dfrac{3 \times 6 + 5 \times 2}{3 + 5} \\[1em] = \dfrac{18 + 10}{8} \\[1em] = \dfrac{28}{8} \\[1em] = \dfrac{7}{2}.

∴ K = (x, 0) = $\Big(\dfrac{7}{2}, 0\Big).$

Hence, the coordinates of K are $\Big(\dfrac{7}{2}, 0\Big)$ and the ratio is 3 : 5.