if the point p (2,1) lies on the line segment joining point a (4,2) B (8,4), then - Maths - Coordinate Geometry

Question

if the point p (2,1) lies on the line segment joining point a
(4,2) & B (8,4), then

AP = 1/3 AB
AP = PB
PB = 1/3 AB
AP = 1/2 AB

Harleen Birdi · Accepted Answer

Dear Student!!

Here is the answer to your question.

We are given point P (2, 1) lies on the line joining points A (4, 2) and B (8, 4).

Using section formula, for finding out the ratio in which point P divides the line segment AB.

$(x, y) = (\frac{m_{1} x_{2} + m_{2} x_{1}}{m_{1} + m_{2}}, \frac{m_{1} y_{2} + m_{2} y_{1}}{m_{1} + m_{2}}) Here, (x, y) = (2, 1) (x_{1}, y_{1}) = (4, 2) (x_{2}, y_{2}) = (8, 4) ∴ (2, 1) = (\frac{m_{1} (8) + m_{2} (4)}{m_{1} + m_{2}}, \frac{m_{1} (4) + m_{2} (2)}{m_{1} + m_{2}}) \Rightarrow 2 = \frac{8 m_{1} + 4 m_{2}}{m_{1} + m_{2}} (\begin{matrix} ∵ if (a, b) = (c, d) \\ \Rightarrow a = c & b = d \end{matrix}) \Rightarrow 1 = \frac{4 m_{1} + 2 m_{2}}{m_{1} + m_{2}} \Rightarrow m_{1} + m_{2} = 4 m_{1} + 2 m_{2} \Rightarrow 3 m_{1} + m_{2} = 0 \Rightarrow m_{2} = - 3 m_{1} \Rightarrow \frac{m_{1}}{m_{2}} = \frac{- 1}{3}$

i.e., m₁ : m₂ = –1 : 3

i.e., point P divides AB in the ratio 1 : 3 externally.

i.e. AP : PB = 1 : 3

So, we can conclude AP : AB = 1 : 2

$\Rightarrow \frac{AP}{AB} = \frac{1}{2} \Rightarrow AP = \frac{1}{2} AB$

Cheers..!!

i e AP : PB = 1 : 3 So, we can conclude AP : AB = 1 : 2 $\Rightarrow \frac{AP}{AB} = \frac{1}{2} \Rightarrow AP = \frac{1}{2} AB$ Cheers !!

if the point p (2,1) lies on the line segment joining point a (4,2) & B (8,4), then AP = 1/3 AB AP = PB PB = 1/3 AB AP = 1/2 AB

if the point p (2,1) lies on the line segment joining point a (4,2) & B (8,4), then

AP = 1/3 AB

AP = PB

PB = 1/3 AB

AP = 1/2 AB