Question

In what ratio does the point \[\left( {\dfrac{{24}}{{11}},y} \right)\], divide the line segment joining the points P\[\left( {2, - 2} \right)\] & Q\[\left( {3,7} \right)\]? Also find the value of y?

Answer 1

Hint: In this question, we have to find out the ratio when we divide a line segment by the given point.
First we put the given points p and Q in the section formula and equate it with the coordinates of the given point then we will get the ratio and the value of y.

Formula used: Section formula:
 The coordinate of P(x, y) which divides the line segment joining the points A \[({x_1},{y_1})\] and B \[({x_2},{y_2})\] internally in the ratio m:n are
\[\left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}} \right)\]

Complete step-by-step answer:
It is given that, the point\[\left( {\dfrac{{24}}{{11}},y} \right)\], divide the line segment joining the points P\[\left( {2, - 2} \right)\] & Q\[\left( {3,7} \right)\].
Let, the point \[\left( {\dfrac{{24}}{{11}},y} \right)\], divide the line segment joining the points P\[\left( {2, - 2} \right)\]& Q\[\left( {3,7} \right)\]in the ratio m:n.
Then applying section formula we get, the coordinate of (x, y) which divides the line segment joining the points P\[\left( {2, - 2} \right)\]& Q\[\left( {3,7} \right)\]internally in the ratio m:n are
\[\left( {\dfrac{{m \times 3 + n \times 2}}{{m + n}},\dfrac{{m \times 7 + n \times - 2}}{{m + n}}} \right)\]
Here (x,y) is given by \[\left( {\dfrac{{24}}{{11}},y} \right)\] .
Equating the x coordinates we get,
$\Rightarrow$\[\dfrac{{m \times 3 + n \times 2}}{{m + n}} = \dfrac{{24}}{{11}}\]
Let us multiply the numerator term and we get,
$\Rightarrow$\[\dfrac{{3m + 2n}}{{m + n}} = \dfrac{{24}}{{11}}\]
Now we have to take cross multiplication and we get
$\Rightarrow$\[33m + 22n = 24m + 24n\]
Let us take same as one side and we get,
$\Rightarrow$\[33m - 24m = 24n - 22n\]
On subtracting we get,
$\Rightarrow$\[9m = 2n\]
Let us divided the term
Then,\[\dfrac{m}{n} = \dfrac{2}{9}\]
Also, equating the y coordinate we get,
$\Rightarrow$ \[\dfrac{{m \times 7 + n \times - 2}}{{m + n}} = y\]
On multiplying the term and we get
$\Rightarrow$\[\dfrac{{7m - 2n}}{{m + n}} = y\]
Dividing numerator and denominator of left hand side by n,
$\Rightarrow$\[\dfrac{{\dfrac{{7m - 2n}}{n}}}{{\dfrac{{m + n}}{n}}} = y\]
Let us rewrite it as,
$\Rightarrow$\[\dfrac{{7\dfrac{m}{n} - 2\dfrac{n}{n}}}{{\dfrac{m}{n} + \dfrac{n}{n}}} = y\]
Putting the value of \[\dfrac{m}{n}\] and we get
$\Rightarrow$\[\dfrac{{7 \times \dfrac{2}{9} - 2}}{{\dfrac{2}{9} + 1}} = y\]
On multiplying and take the LCM of numerator and denominator we get,
$\Rightarrow$\[y = \dfrac{{\dfrac{{14 - 18}}{9}}}{{\dfrac{{2 + 9}}{9}}}\]
On adding we get
$\Rightarrow$\[y = \dfrac{{\dfrac{{ - 4}}{9}}}{{\dfrac{{11}}{9}}}\]
On cancel the denominator term and we get
$\Rightarrow$\[y = \dfrac{{ - 4}}{{11}}\]
Thus we get, \[m:n = 2:9\] and \[y = \dfrac{{ - 4}}{{11}}\].

The point \[\left( {\dfrac{{24}}{{11}},y} \right)\], divide the line segment joining the points P\[\left( {2, - 2} \right)\] & Q\[\left( {3,7} \right)\] in \[2:9\] Also the value of y is \[\dfrac{{ - 4}}{{11}}\].

Note: In geometry, The section formula tells us the coordinates of the point which divides a given line segment into two parts such that their lengths are in the ratio m:n.

The coordinate of P(x,y) which divides the line segment joining the points A \[({x_1},{y_1})\] and B \[({x_2},{y_2})\] internally in the ratio m:n are
\[\left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}} \right)\].