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Question: In what ratio does the point \[\left( {\dfrac{{24}}{{11}},y} \right)\], divide the line segment join...

In what ratio does the point (2411,y)\left( {\dfrac{{24}}{{11}},y} \right), divide the line segment joining the points P(2,−2)\left( {2, - 2} \right) & Q(3,7)\left( {3,7} \right)? Also find the value of y?

Explanation

Solution

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 (x1,y1)({x_1},{y_1}) and B (x2,y2)({x_2},{y_2}) internally in the ratio m:n are
(mx2+nx1m+n,my2+ny1m+n)\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(2411,y)\left( {\dfrac{{24}}{{11}},y} \right), divide the line segment joining the points P(2,−2)\left( {2, - 2} \right) & Q(3,7)\left( {3,7} \right).
Let, the point (2411,y)\left( {\dfrac{{24}}{{11}},y} \right), divide the line segment joining the points P(2,−2)\left( {2, - 2} \right)& Q(3,7)\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(2,−2)\left( {2, - 2} \right)& Q(3,7)\left( {3,7} \right)internally in the ratio m:n are
(m×3+n×2m+n,m×7+n×−2m+n)\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 (2411,y)\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}}$$ Now we have to take cross multiplication and we get $\Rightarrow33m + 22n = 24m + 24n Let us take same as one side and we get, $\Rightarrow$$$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 mn\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, \Rightarrowy = \dfrac{{\dfrac{{14 - 18}}{9}}}{{\dfrac{{2 + 9}}{9}}}$$ On adding we get $\Rightarrowy = \dfrac{{\dfrac{{ - 4}}{9}}}{{\dfrac{{11}}{9}}} On cancel the denominator term and we get $\Rightarrow$$$y = \dfrac{{ - 4}}{{11}}
Thus we get, m:n=2:9m:n = 2:9 and y=−411y = \dfrac{{ - 4}}{{11}}.

The point (2411,y)\left( {\dfrac{{24}}{{11}},y} \right), divide the line segment joining the points P(2,−2)\left( {2, - 2} \right) & Q(3,7)\left( {3,7} \right) in 2:92:9 Also the value of y is −411\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 (x1,y1)({x_1},{y_1}) and B (x2,y2)({x_2},{y_2}) internally in the ratio m:n are
(mx2+nx1m+n,my2+ny1m+n)\left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}} \right).