Question

Find the ratio in which Y-axis divides the point A(3, 5) and point B(– 6, 7). Find the coordinates of the point

Answer 1

Let C be a point on Y-axis which divides seg AB in the ratio m : n

Point C lies on the Y-axis.

∴ its X-coordinate is 0.

Let C = (0, y)

Here,

A(x₁, y₁) = A(3, 5)

B(x₂, y₂) = B(– 6, 7)

By Section formula,

x = ${\displaystyle \frac{\text{m}{x}_2+\text{n}{x}_1}{\text{m}+\text{n}}}$ 

∴ 0 = ${\displaystyle \frac{-6\text{m}+3\text{n}}{\text{m}+\text{n}}}$

∴ – 6m + 3n = 0

∴ 3n = 6m

∴ ${\displaystyle \frac{\text{m}}{\text{n}}=\frac{3}{6}}$

∴ ${\displaystyle \frac{\text{m}}{\text{n}}=\frac{1}{2}}$   ......(i)

∴ m : n = 1 : 2

By section formula,

y =  ${\displaystyle \frac{\text{m}{y}_2+\text{n}{y}_1}{\text{m}+\text{n}}}$ 

y = ${\displaystyle \frac{7\text{m}+5\text{n}}{\text{m}+\text{n}}}$

= ${\displaystyle \frac{7\text{m}+5\left(2\text{m}\right)}{\text{m}+2\text{m}}}$  ......[From (i), n = 2m]

= ${\displaystyle \frac{7\text{m}+10\text{m}}{3\text{m}}}$

= ${\displaystyle \frac{17\text{m}}{3\text{m}}}$

∴ Y-axis divides the seg AB in the ratio 1 : 2 and the co-ordinates of that point is ${\displaystyle (0,\frac{17}{3})}$.