Question

Find the ratio in which the P(4,y) divides the line segment joining the points A(2,3) and B(6,-3).Hence find the value of y. ​

Answer 1

Answer- The above question is from the chapter 'Coordinate Geometry'.

Concept used: 1) Section Formula:-

Let P (x, y) be a point on a line joining A (x₁, y₁) and B (x₂, y₂).

Let it divide AB in the ratio k : 1.

Then,

$x = \dfrac{kx_{2} + x_{1}}{k +1}\\\\y = \dfrac{ky_{2} + y_{1}}{k +1}$

2) Mid point formula:-

Let P (x, y) be the midpoint of a line joining A (x₁, y₁) and B (x₂, y₂).

Then,

$x = \dfrac{x_1 + x_2}{2} \: \: and \: \: y = \dfrac{y_1 + y_2}{2}$

Given question: Find the ratio in which the P(4,y) divides the line segment joining the points A (2,3) and B (6,-3). Hence find the value of y. ​

Solution: Let P (4, y) be a point on a line joining A (2, 3) and B (6, -3) dividing it in the ratio k : 1.

Then,

$4 = \dfrac{6k + 2}{k + 1}\\\\4k + 4 = 6k + 2\\2k = 2\\k = 1\\\\\implies k:1 = 1:1$

⇒ P (4, y) is the midpoint of AB.

Using midpoint formula for y, we get,

$y = \dfrac{3 + (-3)}{2} \\\\y = \dfrac{0}{2} \\\\y = 0$

∴ P (4, y) divides AB in the ratio 1 : 1 and value of y = 0.