Question

Find the ratio in which the point p whose ordinate is -3 divides the join of A(-2,3) and B(5,-15/2).

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Answer 1

Given:

• A point P whose ordinate is -3
• Point P divides the line segment joining point A(-2,3) and point B(5,-15/2)

To Find:

• Ratio in which point P divides the line segment AB

Sectional Formula

$\setlength{\unitlength}{0.9 cm}}\begin{picture}(12,4)\thicklines\put(6,6){\line(1,0){5.5}}\put(5.6,5.9){ P }\put(11.7,5.9){ Q }\put(5.4,5.5){ (x_1\,,\,y_1) }\put(11.4,5.5){ (x_2\,,\,y_2) }\put(8,6){\circle*{0.2}}\put(7.8,6.3){ R }\put(7.4,5.5){ (x\,,\,y) }\put(6.6,6.3){ m }\put(9.3,6.3){ n }\put(11.7,5.9){ Q }\end{picture}$

Let P(x₁,y₁) and Q(x₂,y₂) be two points. Let the point R(x,y) divide the line segment joining the points P and Q internally in the ratio m:n, then

$\sf{(x,y)=\bigg(\dfrac{mx_{2}+nx_{1}}{m+n},\dfrac{my_{2}+ny_{1}}{m+n}\bigg)}$

Solution:

Let the ratio in which point P divides the line segment AB be λ:1

Also, let abscissa(x-coordinate) of point P be 'a'

Diagram:-

$\setlength{\unitlength}{0.9 cm}}\begin{picture}(12,4)\thicklines\put(6,6){\line(1,0){5.5}}\put(5.6,5.9){ A }\put(11.7,5.9){ B }\put(5.4,5.5){ (-2\,,\,3) }\put(11.4,5.5){ (5\,,\,-\dfrac{15}{2} ) }\put(8,6){\circle*{0.2}}\put(7.8,6.3){ P }\put(7.4,5.5){ (a\,,\,-3) }\put(6.6,6.3){ \lambda }\put(9.3,6.3){ 1 }\put(11.7,5.9){ B }\end{picture}$

Now, by using sectional formula, we get

$\longrightarrow\sf{(a,-3)=\bigg(\dfrac{\lambda(5)-2}{\lambda+1},\dfrac{\lambda(\dfrac{-15}{2} ) +3}{\lambda +1}\bigg)}$

$\longrightarrow\sf{(a,-3)=\bigg(\dfrac{5\lambda-2}{\lambda+1},\dfrac{(\dfrac{-15\lambda+6}{2} )}{\lambda +1}\bigg)}$

$\longrightarrow\sf{(a,-3)=\bigg(\dfrac{5\lambda-2}{\lambda+1},\dfrac{-15\lambda+6}{2(\lambda +1)}\bigg)}$

On comparing both the sides we get,

$\rightarrow\sf{\dfrac{-15\lambda+6}{\:\:2(\lambda+1)}=-3}$

$\rightarrow\sf{-15\lambda+6=-6(\lambda+1)}$

$\rightarrow\sf{-15\lambda+6=-6\lambda-6}$

$\rightarrow\sf{6+6=15\lambda-6\lambda}$

$\rightarrow\sf{12=9\lambda}$

$\rightarrow\sf{9\lambda=12}$

$\rightarrow\sf{\lambda=\dfrac{\cancel{12}}{\cancel{9}}}$

$\rightarrow\sf{\lambda=\dfrac{4}{3}}$

So, the required ratio is

$\sf{\dfrac{\lambda}{1}=\dfrac{\frac{4}{3}}{1}}$

$\sf{\dfrac{\lambda}{1}=\dfrac{4}{3}}$

λ:1= 4:3

Hence, point P divides the line segment AB in 4:3.

Answer 2

Step-by-step explanation:

4:3

no faltu answer