Question

In what ratio is the line segment joining the points (-2, -3) and (3, 7) divided by y - axis?
(a) 5 : 1
(b) 1 : 5
(c) 2 : 3
(d) 3 : 2

Answer 1

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 : n

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:

$P=\left( \dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n},\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n} \right)$
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.