Question

Find the ratio in which the point (1, -k) divides the line joining the points (-3, 10) and (6, -8). And find the value of k.

Answer 1

Hint: In this particular question use the concept of section formula if three points are collinear (i.e. they are in same line), so a point (x, y) divides the line joining the points $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ internally in the ratio (m : n) then the coordinates of the point (x, y) is given as, $\left( {x,y} \right) = \left( {\dfrac{{m{x_2} + n{x_1}\,}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}} \right)$ so use this concept to reach the solution of the question.

Complete step by step answer:
Let, A = (-3, 10) = $\left( {{x_1},{y_1}} \right)$
B = (6, -8) = $\left( {{x_2},{y_2}} \right)$
Now, let the point (-1, k) divides the line AB in the ratio (m : n)
Let, C = (-1, k) = (x, y)

Now according to section formula if a point (x, y) divides the line joining the points $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ internally in the ratio (m : n) then the coordinates of the point (x, y) is given as,
$ \Rightarrow \left( {x,y} \right) = \left( {\dfrac{{m{x_2} + n{x_1}\,}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}} \right)$
Now substitute all the values we have,
$ \Rightarrow \left( { - 1,k} \right) = \left( {\dfrac{{m\left( 6 \right) + n\left( { - 3} \right)\,}}{{m + n}},\dfrac{{m\left( { - 8} \right) + n\left( {10} \right)}}{{m + n}}} \right)$
Now simplify we have,
$ \Rightarrow \left( { - 1,k} \right) = \left( {\dfrac{{6m - 3n\,}}{{m + n}},\dfrac{{ - 8m + 10n}}{{m + n}}} \right)$
Now on comparing we have,
$ \Rightarrow - 1 = \dfrac{{6m - 3n\,}}{{m + n}}$................. (1)
And
$k = \dfrac{{ - 8m + 10n}}{{m + n}}$.............. (2)
Now first simplify equation (1) we have,
$ \Rightarrow - 1 = \dfrac{{6m - 3n\,}}{{m + n}}$
$ \Rightarrow - m - n = 6m - 3n\,$
$ \Rightarrow - m - 6m = n - 3n\,$
$ \Rightarrow - 7m = - 2n\,$
$ \Rightarrow \dfrac{m}{n} = \dfrac{2}{7}$................. (3)
So this is the required ratio in which the point (1, -k) divides the line joining the points (-3, 10) and (6, -8).
Now from equation (2) we have,
$ \Rightarrow k = \dfrac{{ - 8m + 10n}}{{m + n}}$
Now divide by (n) in numerator and denominator of the above equation we have,
$ \Rightarrow k = \dfrac{{ - 8\left( {\dfrac{m}{n}} \right) + 10}}{{\left( {\dfrac{m}{n}} \right) + 1}}$
Now from equation (3) we have,
$ \Rightarrow k = \dfrac{{ - 8\left( {\dfrac{2}{7}} \right) + 10}}{{\left( {\dfrac{2}{7}} \right) + 1}}$
Now simplify this we have,
$ \Rightarrow k = \dfrac{{ - 8\left( {\dfrac{2}{7}} \right) + 10}}{{\left( {\dfrac{2}{7}} \right) + 1}} = \dfrac{{ - 16 + 70}}{{2 + 7}} = \dfrac{{54}}{9} = 6$

So the required value of k is 6.

Note: Whenever we face such types of questions the key concept we have to remember is the section formula which is stated above, then simply substitute the values in this formula and simplify as above we will get the required ratio and the value of k, which is our required answer.