In what ratio is the line segment joining the points (2, -3) and (5, 6)

The exam dates for the HTET 2022 have been postponed. Due to the General Elections, the exam dates for the HTET have been revised. The exam will be conducted on the 3rd and 4th of December 2022 instead of the 12th and 13th of November 2022. The exam is conducted by the Board of School Education, Haryana to shortlist eligible candidates for PGT and TGT posts in Government schools across Haryana. The exam is conducted for 150 marks. The HTET Exam Pattern for Level I, Level II, and Level III exams is different. There will be no negative marking in the exam.

Answer

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.

In what ratio is the line joining (2, -3) and (5, 6) divided by the x-axis.

Let the line joining points A (2, −3) and B (5, 6) be divided by point P (x, 0) in the ratio k : 1.

`y=(ky_2+y_1)/(k+1)`

`0=(kxx6+1xx(-3))/(k+1)`

`0=6k-3`

`k=1/2`

Thus, the required ratio is 1: 2.

Concept: Co-ordinates Expressed as (x,y)

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Page 2

In what ratio is the line joining (2, -4) and (-3, 6) divided by the y – axis.

Let the line joining points A (2, -4) and B (-3, 6) be divided by point P (0, y) in the ratio k : 1.

`x=(kx_2+x_1)/(k+1)`

`0=(kxx(-3)+1xx2)/(k+1)`

`0=-3k+2`

`k=2/3`

Thus, the required ratio is 2: 3.

Concept: Co-ordinates Expressed as (x,y)

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Page 3

In what ratio does the point (1, a) divide the join of (-1, 4) and (4,-1)? Also, find the value of a.

Let the point P (1, a) divides the line segment AB in the ratio k: 1. 
Using section formula, we have:

`1=(4k-1)/(k+1)`

`=>k+1=4k-1`

`=>2=3k`

`=>k=2/3`  ............(1)

`=>a=(-k+4)/(k+1)`

`=> a = (-2/3 + 4)/(2/3 + 1)`     (from 1)

`=> a = 10/5 = 2`

Hence, the required is 2 : 3 and the value of a is 2.

Concept: Co-ordinates Expressed as (x,y)

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Page 4

In what ratio does the point (a, 6) divide the join of (-4, 3) and (2, 8)? Also, find the value of a.

Let the point P (a, 6) divides the line segment joining A (-4, 3) and B (2, 8) in the ratio k: 1.
Using section formula, we have:

`6=(8k+3)/(k+1)`

`=> 6k+6=8k+3`

`=>3=2k`

`=>k=3/2`  .................(1)

`=>a=(2k-4)/(k+1)`

`=>a=(2xx3/2-4)/(3/2+1)`   (from equation 1)

`=>a=-2/5`

Hence, the required ratio is 3:2 and the value of a is `-2/5`

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