Find the ratio in which the line segment joining the points (5, 3)

Q) Find the ratio in which the line segment joining the points (5, 3) and (–1, 6) is divided by Y-axis.

Ans:

Step 1: By section formula, coordinates of point P (X, Y) which lies between two points (x₁, y₁), (x₂, y₂) will be given by:

P (X,Y) = $(\frac{m_1 \times_2 + m_2 \times_1}{m_1 + m_2}, \frac{m_1 Y_2 + m_2 Y_1}{m_1 + m_2})$

here, point divides the line in ratio of m_{1 :} m₂

Now if the given points A (5, 3) and B(- 1, 6) are divided in the ratio of m : n, then:

coordinates of dividing point (x, y) = $(\frac{m (- 1) + n (5)}{m + n}, \frac{m (6) + n (3)}{m + n})$

= $(\frac{(- m + 5 n)}{m + n}, \frac{(6 m + 3 n)}{m + n})$

Next, since this point lies on line Y axis, it means x = 0, this point will satisfy the equation

∴ $\frac{(- m + 5 n)}{m + n}$ = 0

∴ – m + 5 n = 0

∴ m = 5 n

∴ m : n = 5 : 1

Therefore, the line segment divides the line in ratio of 5:1.

Check: if we plot the given points and connect them:

Here, we can see that Y axis is cutting them in a ratio where m is considerably larger than n. hence our answer is correct.

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