In what ratio is the line joining the points (- 3 2 and 61 is divided by Y axis?

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.

The ratio in which the y-axis divides two points (x1 , y1)  and  (x2 , y2)  is  \[\lambda: 1\] 

The co-ordinates of the point dividing two points (x1 , y1)  and (x2 , y2)   in the ratio m : n  is given as,

`(x , y) = ((lambdax_2 + x_1)/(lambda + 1 )) ,((lambday_2 + y_1)/(lamda + 1))`  where, `lambda = m/n`

Here the two given points are A(5,−6) and B(−1,−4).

\[(x, y) = \left( \frac{- \lambda + 5}{\lambda + 1}, \frac{- 4\lambda - 6}{\lambda + 1} \right)\]

Since, the y-axis divided the given line, so the x coordinate will be 0.

\[\frac{- \lambda + 5}{\lambda + 1} = 0\]
\[\lambda = \frac{5}{1}\]

Thus the given points are divided by the y-axis in the ratio  5:1.

The co-ordinates of this point (x, y) can be found by using the earlier mentioned formula.

`(x , y ) = ((5/1 (-1) + (5) )/(5/1 + 1)) , ((5/1(-4)+(-6))/(5/1 +1))`

`(x , y) = (0/6) , (-26/6)`

`(x , y )  = ( 0 , - 26/6)`

Thus the co-ordinates of the point which divides the given points in the required ratio are `(0,-26/6)`.

Solution:

The coordinates of the point P(x, y) which divides the line segment joining the points A(x₁, y₁) and B(x₂, y₂), internally, in the ratio m₁: m₂ is given by the Section Formula: P(x, y) = [(mx₂ + nx₁) / m + n, (my₂ + ny₁) / m + n]

Let the ratio in which the line segment joining A(- 3, 10) and B(6, - 8) be divided by point C(- 1, 6) be k : 1.

By Section formula, C(x, y) = [(mx₂ + nx₁) / m + n, (my₂ + ny₁) / m + n]

m = k, n = 1

Therefore,

- 1 = (6k - 3) / (k + 1)

- k - 1 = 6k - 3

7k = 2

k = 2 / 7

Hence, the point C divides line segment AB in the ratio 2 : 7.

☛ Check: NCERT Solutions for Class 10 Maths Chapter 7

Video Solution:

NCERT Class 10 Maths Solutions Chapter 7 Exercise 7.2 Question 4

Summary:

The ratio in which the line segment joining the points (- 3, 10) and (6, - 8) is divided by (- 1, 6) is 2 : 7.

☛ Related Questions:

• Find the ratio in which the line segment joining A (1, - 5) and B (- 4, 5) is divided by the x-axis. Also, find the coordinates of the point of division.
• If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y.
• Find the coordinates of a point A, where AB is the diameter of a circle whose centre is (2, - 3) and B is (1, 4).
• If A and B are (- 2, - 2) and (2, - 4), respectively, find the coordinates of P such that AP = 3/7 AB and P lies on the line segment AB.

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