Question

Find the coordinates of the point on the line joining \[P(1,-2)\] and \[Q(4,7)\] that is twice as far from P as from Q?

Answer 1

Hint: From the question, it was given that the point on the line joining \[P(1,-2)\] and \[Q(4,7)\] that is twice as far from P as from Q. Now we have to find the ratio such that the point divides the line joining \[P(1,-2)\] and \[Q(4,7)\]. Let us assume a point \[R(x,y)\] divides the line joining \[P(1,-2)\] and \[Q(4,7)\] according to the ratio given in the question. We should know that if \[P({{x}_{1}},{{y}_{1}})\] and \[Q({{x}_{2}},{{y}_{2}})\] are divided by \[R(x,y)\] in the ratio \[m:n\] then \[x=\dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n}\] and \[y=\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n}\]. Now we should find the coordinates of the point \[R(x,y)\].

Complete step-by-step answer:
Before solving the question, we should know that if \[P({{x}_{1}},{{y}_{1}})\] and \[Q({{x}_{2}},{{y}_{2}})\]are divided by \[R(x,y)\]in the ratio \[m:n\]then \[x=\dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n}\] and \[y=\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n}\].
From the question, a point on the line joining \[P(1,-2)\] and \[Q(4,7)\] that is twice as far from P as from Q.

Let us assume a point \[R(x,y)\] divides the line joining \[P(1,-2)\] and \[Q(4,7)\] in the ratio \[2:1\].
Now let us compare \[P(1,-2)\] with \[P({{x}_{1}},{{y}_{1}})\], then we get
\[\begin{align}
  & {{x}_{1}}=1.....(1) \\
 & {{y}_{1}}=-2.....(2) \\
\end{align}\]
Now let us compare \[Q(4,7)\] with \[Q({{x}_{2}},{{y}_{2}})\], then we get
\[\begin{align}
  & {{x}_{2}}=4.....(3) \\
 & {{y}_{2}}=7.....(4) \\
\end{align}\]
We should know that if \[P({{x}_{1}},{{y}_{1}})\] and \[Q({{x}_{2}},{{y}_{2}})\]are divided by \[R(x,y)\]in the ratio \[m:n\]then \[x=\dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n}\] and \[y=\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n}\].
In the similar way, if \[P(1,-2)\] and \[Q(4,7)\] are divided by \[R(x,y)\] in the ratio \[m:n\]then \[x=\dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n}\] and \[y=\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n}\].
We know that the \[R(x,y)\] divides \[P(1,-2)\] and \[Q(4,7)\] in the ratio \[2:1\].
Now let us compare \[m:n\]with \[2:1\], then we get
\[\begin{align}
  & m=2.....(5) \\
 & n=1.......(6) \\
\end{align}\]
Now by using equation (1), equation (2), equation (3), equation (4), equation (5) and equation (6), we should find the coordinates of \[R(x,y)\].
Let us find the x-coordinate of \[R(x,y)\].
\[\begin{align}
  & x=\dfrac{(2)(4)+(1)(1)}{2+1} \\
 & \Rightarrow x=\dfrac{9}{3} \\
 & \Rightarrow x=3.....(7) \\
\end{align}\]
Let us find the y-coordinate of \[R(x,y)\].
\[\begin{align}
  & y=\dfrac{(2)(7)+(1)(-2)}{2+1} \\
 & \Rightarrow y=\dfrac{12}{3} \\
 & \Rightarrow y=4.....(8) \\
\end{align}\]

From equation (7) and equation (8), it is clear that the value of x is equal to 2 and the value of y is equal to 1.
So, it is clear that if \[P(1,-2)\] and \[Q(4,7)\] are divided by \[R(x,y)\] in the ratio \[2:1\] then \[x=3\] and \[y=4\].
So, \[R\left( 3,4 \right)\] divides \[P(1,-2)\] and \[Q(4,7)\] in the ratio \[2:1\].

Note: Students may have a misconception that a point \[R(x,y)\] divides the line joining \[P(-1,2)\] and \[Q(4,7)\] in the ratio \[1:2\]. Then the solution is as follows.
Now let us compare \[P(-1,2)\] with \[P({{x}_{1}},{{y}_{1}})\], then we get
\[\begin{align}
  & {{x}_{1}}=-1.....(1) \\
 & {{y}_{1}}=2.....(2) \\
\end{align}\]
Now let us compare \[Q(4,7)\] with \[Q({{x}_{2}},{{y}_{2}})\], then we get
\[\begin{align}
  & {{x}_{2}}=4.....(3) \\
 & {{y}_{2}}=7.....(4) \\
\end{align}\]
We should know that if \[P({{x}_{1}},{{y}_{1}})\] and \[Q({{x}_{2}},{{y}_{2}})\]are divided by \[R(x,y)\]in the ratio \[m:n\]then \[x=\dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n}\] and \[y=\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n}\].
In the similar way, if \[P(-1,2)\] and \[Q(4,7)\] are divided by \[R(x,y)\] in the ratio \[m:n\]then \[x=\dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n}\] and \[y=\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n}\].
We know that the \[R(x,y)\] divides \[P(-1,2)\] and \[Q(4,7)\] in the ratio \[1:2\].
Now let us compare \[m:n\]with \[1:2\], then we get
\[\begin{align}
  & m=1.....(5) \\
 & n=2.......(6) \\
\end{align}\]
Now by using equation (1), equation (2), equation (3), equation (4), equation (5) and equation (6), we should find the coordinates of \[R(x,y)\].
Let us find the x-coordinate of \[R(x,y)\].
\[\begin{align}
  & x=\dfrac{(1)(4)+(2)(-1)}{1+2} \\
 & \Rightarrow x=\dfrac{2}{3}.....(7) \\
\end{align}\]
Let us find the y-coordinate of \[R(x,y)\].
\[\begin{align}
  & y=\dfrac{(1)(7)+(2)(2)}{1+2} \\
 & \Rightarrow y=\dfrac{11}{3}.....(8) \\
\end{align}\]
From equation (7) and equation (8), it is clear that the value of x is equal to \[\dfrac{2}{3}\] and the value of y is equal to \[\dfrac{11}{3}\].
So, it is clear that if \[P(-1,2)\] and \[Q(4,7)\] are divided by \[R(x,y)\] in the ratio \[1:2\] then \[x=\dfrac{2}{3}\] and \[y=\dfrac{11}{3}\].
So, \[R\left( \dfrac{2}{3},\dfrac{11}{3} \right)\] divides \[P(-1,2)\] and \[Q(4,7)\] in the ratio \[1:2\].
So, we know the required point is \[R\left( 3,4 \right)\]. Hence, \[R\left( \dfrac{2}{3},\dfrac{11}{3} \right)\] is the wrong answer.