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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?

Answer
VerifiedVerified
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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 \[{{x}_{1}},\,{{x}_{2}},\,{{y}_{1}}\,and\,{{y}_{2}}\].