Question

A line AB is drawn on a plane. Points P and Q are marked on the line AB on the same side of mid point of AB. P divides the line AB in the ratio 2 : 3, and Q divides the line AB in the ratio 3 : 4. If PQ is given to be 2 then find the length of AB.

Answer 1

Hint: First draw the diagram with the help of given information in the question. Then write the ratio of length segments formed by P and Q. Then form an equation involving PQ. Then put PQ = 2 and solve the equation.

Complete step-by-step answer:

Let’s draw a line AB and mark points P and Q on it such that P and Q are on the same side of mid point.
It is given that P divides the line segment AB in the ratio 2 : 3. So
$\dfrac{AP}{PB}=\dfrac{2}{3}\,\,\,\,\,\,\,\,\,\cdot \cdot \cdot \text{(i)}$
It is also given that Q divides the line segment AB in the ratio 3 : 4. So,
$\begin{align}
  & \dfrac{AQ}{QB}=\dfrac{3}{4}\, \\
 & \Rightarrow AQ=\dfrac{3}{4}QB\,\,\,\,\,\,\cdot \cdot \cdot \text{(ii)} \\
\end{align}$
Taking equation (i),
$\begin{align}
  & \dfrac{AP}{PB}=\dfrac{2}{3} \\
 & \Rightarrow \dfrac{AQ-PQ}{PQ+QB}=\dfrac{2}{3} \\
 & \Rightarrow 3\left( AQ-QB \right)=2\left( PQ+QB \right) \\
\end{align}$
PQ is given to be 2. So putting it in the above equation we get
$\begin{align}
  & 3\left( AQ-PQ \right)=2\left( PQ+QB \right) \\
 & \Rightarrow 3(AQ-2)=2(2+QB) \\
 & \Rightarrow 3AQ-6=4+2QB \\
 & \Rightarrow 3AQ-2QB=10\,\,\,\,\,\cdot \cdot \cdot \text{(iii)} \\
\end{align}$
Now using equation (ii) in the equation (iii) we get
$\begin{align}
  & 3\dfrac{3}{4}QB-2QB=10 \\
 & \Rightarrow \dfrac{1}{4}QB=10 \\
 & \Rightarrow QB=40 \\
\end{align}$
Putting this value of QB in equation (ii) we get,
$\begin{align}
  & AQ=\dfrac{3}{4}QB \\
 & \Rightarrow AQ=\dfrac{3}{4}\times 40=30 \\
\end{align}$
Now we know the values of AQ and QB. To get the value of AB we have to add AQ and QB. So,
$\begin{align}
  & AB=AQ+QB \\
 & \Rightarrow AB=30+40 \\
 & \Rightarrow AB=70 \\
\end{align}$
So the length of line segment AB is 70.
Note: This question requires making a diagram to best visualise the question. If the figure is not drawn then you may make mistakes in substituting AP and PB.