Question

If point P(1/2 ,y ) lies on line segment joining the points A(3 -5) and B(-7,9) then find the ratio in which P divides AB also find the value of y

Answer 1

I think it's clear.

Answer 2

Answer: The ratio is 1:3 and  P is $(\dfrac{1}{2} ,\dfrac{-3}{2} )$

Step-by-step explanation:

$A(3,-5) ,B(-7,9)\text { is a line segment and }P(\dfrac{1}{2} ,y)$ lies the line segment

Let P divides the line segment in k:1

Therefore as we know section formula

$(x,y)= (\dfrac{m_1x_2+m_2x_1}{m_1+m_2} ,\dfrac{m_1y_2+m_2y_1}{m_1+m_2} )$

Now

$A(3,-5)\text { is } (x_1,y_1), B(-7,9)\text { is }(x_2,y_2)\text { and }m_1:m_2\text { is }k:1$

corresponding to x

$\dfrac{1}{2} = \dfrac{k\times-7+1\times 3 }{k+1} \\\\\Rightarrow k+1= 2(-7k+3)\\\\\Rightarrow k+1= -14k+6\\\\\Rightarow 15k=5\\\\\Rightarrow k= \dfrac{1}{3}$

Now

$y= \dfrac{1\times 9+3\times (-5)}{3+1} = \dfrac{9-15}{4} =\dfrac{-6}{4} =\dfrac{-3}{2}$

Hence, the ratio is 1:3 and the coordinate of P is $(\dfrac{1}{2} ,\dfrac{-3}{2} )$