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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.

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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