Question

Calculate the ratio in which the line joining A(-4,2) and B(-3,6) is divided by the point P(x,3). Also, find the length of AP

Answer 1

Hint-Firstly use the section formula and try to find out the coordinates of P and then using the distance formula, find the length of AP

Complete step-by-step answer:
Let us consider that the point P(x,3) divides the line segment joining the points A(-4,2) and B(3,6) in the ratio k:1
Here ${x_1} = - 4,{x_2} = 3;{y_1} = 2,{y_2} = 6$ and ${m_1} = k,{m_2} = 1$
Now , let us make use of the coordinate formula
$$ $$ $\left( {\dfrac{{{m_1}{x_2} + {m_2}{x_1}}}{{{m_1} + {m_2}}},\dfrac{{{m_1}{y_2} + {m_2}{y_1}}}{{{m_1} + {m_2}}}} \right)$
So, on substituting the values, we get
$\left( {\dfrac{{3k - 4}}{{k + 1}},\dfrac{{6k + 2}}{{k + 1}}} \right)$
But, the coordinates of P is (x,3)
So, we get
$\dfrac{{3k - 4}}{{k + 1}} = x - - - - - - - - - (i)$
$\dfrac{{6k + 2}}{{k + 1}} = 3$
6k+2=3k+3
->3k=1$ \Rightarrow k = \dfrac{1}{3}$
The ratio was given as equal to k:1, so it would be $\dfrac{1}{3}:1$ =(1:3) internally
Put the value of k in (i),so we get
$
  \dfrac{{3 \times \dfrac{1}{3} - 4}}{{\dfrac{1}{3} + 1}} = x \\
   \Rightarrow \dfrac{{ - 3}}{{\dfrac{4}{3}}} = x \\
   \Rightarrow \dfrac{{ - 9}}{4} = x \\
 $
So, from this we get the value of x=$\dfrac{{ - 9}}{4}$
So, the coordinates of the point P would be
$\left( {\dfrac{{ - 9}}{4},3} \right)$
Now , we also have to find out the distance between AP
So, for this let us apply the distance formula and find out the value
We have A=(-4,2) and P=$\left( {\dfrac{{ - 9}}{4},3} \right)$
We know the distance formula $d = \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2}} $
On substituting the values, we get
\[
  d = \sqrt {{{\left( {\dfrac{{ - 9}}{4} + 4} \right)}^2} + {{(3 - 2)}^2}} \\
   = \sqrt {{{\left( {\dfrac{{ - 9 + 16}}{4}} \right)}^2} + {1^2}} = \sqrt {\dfrac{{49}}{{16}} + 1} \\
   = \sqrt {\dfrac{{49 + 16}}{{16}}} = \sqrt {\dfrac{{65}}{{16}}} = \dfrac{{\sqrt {65} }}{4} \\
 \]
So, therefore the length of AP= $\dfrac{{\sqrt {65} }}{4}$ units
Note: When solving these type of questions, apply the appropriate formulas wherever needed and also give importance to the type of division in the question(that is if it is internal or external) and try to solve it