In what ratio is the line segment joining the points 4 5 and 1/2 divided by the Y axis find also the co ordinates 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)`.

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. Maths Q&A

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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 be k : 1

Let the line segment be AB joining A (1, - 5) and B (- 4, 5)

By using the Section formula,

P (x, y) = [(mx₂ + nx₁) / m + n, (my₂ + ny₁) / m + n]

m = k, n = 1

Therefore, the coordinates of the point of division is

(x, 0) = [(- 4k + 1) / (k + 1), (5k - 5) / (k + 1)] ---------- (1)

We know that y-coordinate of any point on x-axis is 0.

Therefore, (5k - 5) / (k + 1) = 0

5k = 5

k = 1

Therefore, the x-axis divides the line segment in the ratio of 1 : 1.

To find the coordinates let's substitute the value of k in equation(1)

Required point = [(- 4(1) + 1) / (1 + 1), (5(1) - 5) / (1 + 1)]

= [(- 4 + 1) / 2, (5 - 5) / 2]

= [- 3/2, 0]

☛ Check: NCERT Solutions for Class 10 Maths Chapter 7

Video Solution:

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

NCERT Class 10 Maths Solutions Chapter 7 Exercise 7.2 Question 5

Summary:

The ratio in which the line segment joining A (1, - 5) and B (- 4, 5) is divided by the x-axis is 1:1 and the coordinates of the point of division is (-3/2, 0).

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