Question

Given O(0,0), P(1,2) and S(-3,0). P divides OQ in the ratio 2:3 and OPRS is a parallelogram. Find the coordinates of Q.

Answer 1

Hint: First, proceeding for this, we must know that P is dividing the line segment OQ in the ratio 2:3. Then, we also know the formula named as section formula for the line with end points $A\left( {{x}_{1}},{{y}_{1}} \right)$and $B\left( {{x}_{2}},{{y}_{2}} \right)$ with the intersection point $C\left( {{x}_{3}},{{y}_{3}} \right)$ and ratio of dividing as m:n, then we get${{x}_{3}}=\dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n}$ and ${{y}_{3}}=\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n}$. Then, substituting the values of ${{x}_{1}}$as 0, ${{x}_{3}}$as 1, m as 2, n as 3 and ${{x}_{2}}$as x and by substituting the values of ${{y}_{1}}$as 0, ${{y}_{3}}$as 2, m as 2, n as 3 and ${{y}_{2}}$as y, we get the required values.

Complete step by step answer:
In this question, we are supposed to find the coordinates of Q when O(0,0), P(1,2), and S(-3,0). P divides OQ in the ratio 2:3 and OPRS is a parallelogram.

So, before proceeding for this, we must know that P is dividing the line segment OQ in the ratio 2:3.
Now, let us suppose the coordinates of the point Q as (x,y).
Then, we also know the formula named as section formula for the line with end points $A\left( {{x}_{1}},{{y}_{1}} \right)$and $B\left( {{x}_{2}},{{y}_{2}} \right)$ with the intersection point $C\left( {{x}_{3}},{{y}_{3}} \right)$ and ratio of dividing as m:n, then we get:
${{x}_{3}}=\dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n}$ and ${{y}_{3}}=\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n}$
Then, by substituting the values of ${{x}_{1}}$as 0, ${{x}_{3}}$as 1, m as 2, n as 3 and ${{x}_{2}}$as x, we get the expression as:
$1=\dfrac{2x+3\left( 0 \right)}{2+3}$
Now, by solving the above expression, we get:
$\begin{align}
  & 1=\dfrac{2x+0}{5} \\
 & \Rightarrow 5=2x \\
 & \Rightarrow x=\dfrac{5}{2} \\
\end{align}$
Then, by substituting the values of ${{y}_{1}}$as 0, ${{y}_{3}}$as 2, m as 2, n as 3 and ${{y}_{2}}$as y, we get the expression as:
$2=\dfrac{2y+3\left( 0 \right)}{2+3}$
Now, by solving the above expression, we get:
$\begin{align}
  & 2=\dfrac{2y+0}{5} \\
 & \Rightarrow 10=2y \\
 & \Rightarrow y=\dfrac{10}{2} \\
 & \Rightarrow y=5 \\
\end{align}$
So, we get the coordinates of Q as $\left( \dfrac{5}{2},5 \right)$.
Hence, $\left( \dfrac{5}{2},5 \right)$is the required answer.
Note:
Now, to solve these type of the questions we need to know some of the basic of the formula for the section of the line segments. So, the section formula for the line segment with ratio as m:n with end points $A\left( {{x}_{1}},{{y}_{1}} \right)$and $B\left( {{x}_{2}},{{y}_{2}} \right)$ with the intersection point $C\left( {{x}_{3}},{{y}_{3}} \right)$, then we get:
${{x}_{3}}=\dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n}$ and ${{y}_{3}}=\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n}$