Question

Find the ratio in which the points \[\left( {2,y} \right)\] divides the line segment joining the points \[A\left( { - 2,2} \right)\] and \[B\left( {3,7} \right)\] . Also find the value of y.

Answer 1

Hint: We are given with a point that divides a line segment in a ratio. We will consider the ratio as \[m:n = b:1\] . That it divides the line segment in the ratio so considered. Then in order to find the value of y we will use section formula. Then substituting the values we will get the value of y. we are given the point that divides the line. Remember that!

Complete step by step solution:
Let the point divides the line segment in the ratio \[m:n = b:1\]
Let the point be \[\left( {x,y} \right) = \left( {2,y} \right)\]
Then the section formula is given by,
 \[x = \dfrac{{m{x_2} + n{x_1}}}{{m + n}}\] and \[y = \dfrac{{m{y_2} + n{y_1}}}{{m + n}}\]
Now let the points joining the line segment as \[\left( {{x_1},{y_1}} \right) = A\left( { - 2,2} \right)\& \left( {{x_2},{y_2}} \right) = B\left( {3,7} \right)\]
Putting the values in the formula,
 \[x = \dfrac{{m{x_2} + n{x_1}}}{{m + n}}\]
 \[2 = \dfrac{{b \times 3 + 1 \times \left( { - 2} \right)}}{{b + 1}}\]
On solving the ratio we get,
 \[2 = \dfrac{{3b - 2}}{{b + 1}}\]
Taking the denominator on other side,
 \[2b + 2 = 3b - 2\]
Taking the variables on one side we get,
 \[2 + 2 = 3b - 2b\]
On solving we get,
 \[4 = b\]
Therefore the ratio is 4:1.
 \[y = \dfrac{{m{y_2} + n{y_1}}}{{m + n}}\]
 \[y = \dfrac{{b \times 7 + 1 \times 2}}{{b + 1}}\]
On solving the ratio we get,
 \[y = \dfrac{{7b + 2}}{{b + 1}}\]
Now taking the value of b and putting it in the ratio above,
 \[y = \dfrac{{7 \times 4 + 2}}{{4 + 1}}\]
On solving we get,
 \[y = \dfrac{{30}}{5}\]
On dividing we get,
 \[y = 6\]
This is the value of y.
So, the correct answer is “ \[y = 6\] ”.

Note: Note that why we have taken the ratio of division as \[b:1\] is because 1 will help us in getting the value of b and then putting it into another ratio formula will give the value of y. Also note that if the ratio has both the unknowns then we will definitely get the equations but having both the terms in unknown form. So do take the ratio as \[b:1\] or \[1:b\] .