Question

# In what ratio does the point $\left( \frac{24}{11},\,y \right)$ divides the line segment joining the points P (2, –2) and Q (3, 7) ? Also find the value of y?

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Hint: To find the ratio and the value of y, we have to assume that the line segment PQ is in the ratio of $\left( \lambda :1 \right)$

Complete step by step solution: It is given:
Point $B\left( \frac{24}{11},\,y \right),\,P(2,-2),\,Q(3,7)$
So, \begin{align} & {{x}_{1}}=2,\,{{y}_{1}}=-2 \\ & {{x}_{2}}=3,\,{{y}_{2}}=7 \\ \end{align}
According to the assumption, we can say that
$\left( x=\frac{\lambda {{x}_{2}}+{{x}_{1}}}{\lambda +1} \right)$ ....(1)
$\left( y=\frac{\lambda {{y}_{2}}+{{y}_{1}}}{\lambda +1} \right)$ ....(2)
By putting values and solving the equation (1), we get
\begin{align} & \frac{24}{11}=\frac{\lambda (3)+2}{\lambda +1} \\ & 24\lambda +24=33\lambda +22 \\ & \lambda =\frac{2}{9} \\ \end{align}
By putting values and solving the equation (2), we get
So, the ratio in which B divides PQ =2:9 and the value of y=-(4/11)
\begin{align} & y=\frac{(2/9\times 7)+(-2)}{(2/9+1)} \\ & y=\frac{(14/9)-2}{(11/9)} \\ & y =-\frac{4}{11} \\ \end{align}.

Note: Values should be put correctly. Before putting values, it is important to identify the values of ${{x}_{1}},\,{{x}_{2}},\,{{y}_{1}}\,and\,{{y}_{2}}$.