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

Last updated date: 14th Mar 2023
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Hint: If a point C divides the line joining $A\left( {{x}_{1}},{{y}_{1}} \right)$ and $B\left( {{x}_{2}},{{y}_{2}} \right)$ divides it in the ratio $m:n$, then the coordinates of C are given as : $C=\left( \dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n},\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n} \right)$. Apply this formula to the given condition in the question.

Given points are $p\left( 2,-2 \right)$and $Q\left( 3,7 \right)$ let us assume the point $\left[ \dfrac{24}{11},y \right]$ divides the line segment PQ in the ratio $m:n$.

If a point C divides the line joining $A\left( {{x}_{1}},{{y}_{1}} \right)$ and $B\left( {{x}_{2}},{{y}_{2}} \right)$ divides it in the ratio $m:n$, then the coordinates of C are given as:
$C=\left( \dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n},\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n} \right)$ this is called section formula.

Now, let us apply this section formula for the data given:
$\left( \dfrac{24}{11},y \right)=\left( \dfrac{m\left( 3 \right)+n\left( 2 \right)}{m+n},\dfrac{m\left( 7 \right)+n\left( -2 \right)}{m+n} \right)..........(1)$
Equating the x coordinate from the above equation, we will have: $\dfrac{24}{11}=\dfrac{m\left( 3 \right)+n\left( 2 \right)}{m+n}$.
By cross multiplying we will have: $24\left( m+n \right)=11\left( m\left( 3 \right)+n\left( 2 \right) \right)$
$24m+24n=33m+22n$.
$2n=9m$.
$\dfrac{m}{n}=\dfrac{2}{9}$.
So, we can further conclude that the ratio $m:n$ is nothing but $2:9$.

Now to determine the value of y, let us equate the y -coordinates in equation (1).
$y=\dfrac{m\left( 7 \right)+n\left( -2 \right)}{m+n}$.
As $m=2$and $n=9$, substitute in the above equation we will have:
$y=\dfrac{2\left( 7 \right)+9\left( -2 \right)}{2+9}$
Upon solving,
$y=\dfrac{-4}{11}$.
So, the value of y is $\dfrac{-4}{11}$.
Thus the ratio of $m:n$is $2:9$, and the value of y is $\dfrac{-4}{11}$.
So, the answer we get Ratio is $2:9$ and $y=\dfrac{-4}{11}$

Note: if C divides the joining of two points $A\left( {{x}_{1}},{{y}_{1}} \right)$ and $B\left( {{x}_{2}},{{y}_{2}} \right)$ externally, then the coordinates of C are given as $C=\left( \dfrac{m{{x}_{2}}-n{{x}_{1}}}{m-n},\dfrac{m{{y}_{2}}-n{{y}_{1}}}{m-n} \right)$. As we just have to find the ratio $m:n$, we just need one equation to find out the answer, even if there are two variables in the expression