In what ratio is the line segment joining the points 2 and minus 3 and 3 and 7 divided by the Y axis also find the coordinates of the point of division?

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Hint: First, let the required ratio in which the y axis divides the line joining the points (-2, -3) and (3, 7) be k : 1, and the point of intersection of this line to the y axis to be (0, y). Then use the section formula: If point P(x, y) lies on line segment AB and satisfies AP : PB = m : n, then we say that P divides AB internally in the ratio m : n. The point of division has the coordinates:$P=\left( \dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n},\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n} \right)$. Then find the value of k which is your final answer.Complete step-by-step answer:In this question, we need to find the ratio in which the y axis divides the line joining the points (-2, -3) and (3, 7).Let the required ratio in which the y axis divides the line joining the points (-2, -3) and (3, 7) be k : 1.We will take the point of intersection of this line to the y axis to be (0, y).We will now use the section formula.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 : nIf point P(x, y) lies on line segment AB and satisfies AP : PB = m : n, then we say that P divides AB internally in the ratio m : n. The point of division has the coordinates:$P=\left( \dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n},\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n} \right)$In our question, we have the following:$m=k,n=1,{{x}_{1}}=-2,{{y}_{1}}=-3,{{x}_{2}}=3,{{y}_{2}}=7$We know that the x coordinate of the point of division is 0. Using this, we get the following:$0=\dfrac{k\times 3+1\times \left( -2 \right)}{k+1}$$0=3k-2$$3k=2$ $k=\dfrac{2}{3}$ So, the required ratio in which the y axis divides the line joining the points (-2, -3) and (3, 7) is 2 : 3.Hence, option (c) is correct.Note: In this question, it is very important to let the required ratio in which the y axis divides the line joining the points (-2, -3) and (3, 7) be k : 1, and the point of intersection of this line to the y axis to be (0, y). It is also important to know that if point P(x, y) lies on line segment AB and satisfies AP : PB = m : n, then we say that P divides AB internally in the ratio m : n. The point of division has the coordinates: $P=\left( \dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n},\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n} \right)$. We took the ratio as k:1 to reduce the number of variables.

i In what ratio is the line segment joining the points 2, 3 and 3, 7 divided by the y axis ? Also, find the coordinates of the point of division. ii In what ratio is the line segment joining 3, 1 and 8, 9 divided at the point 5, 21/5 ?

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Let AB be divided by the x-axis in the ratio :1 k at the point P.

Then, by section formula the coordination of P are

`p = ((3k-2)/(k+1) , (7k-3)/(k+1))`

But P lies on the y-axis; so, its abscissa is 0.
Therefore , `(3k-2)/(k+1) = 0`

`⇒ 3k-2 = 0 ⇒3k=2 ⇒ k = 2/3 ⇒ k = 2/3 `

Therefore, the required ratio is `2/3:1`which is same as 2 : 3
Thus, the x-axis divides the line AB in the ratio 2 : 3 at the point P.

Applying `k= 2/3,` we get the coordinates of point.

`p (0,(7k-3)/(k+1))`

`= p(0, (7xx2/3-3)/(2/3+1))`

`= p(0, ((14-9)/3)/((2+3)/3))`