**Q) In what ratio does the X-axis divide the line segment joining the points(2, – 3) and (5, 6)? Also, find the coordinates of the point of intersection.
**

**Ans: **Let’s draw the diagram to solve:

**(i) Ratio of division:**

We know that the value of y-coordinate on X- axis is always 0

It is given that X-axis intersects the line connecting points P &Q, hence the value of y-coordinate of intersection point A will be 0.

∴ we can consider the coordinates of point A as (x, 0)

Next, let’s consider that the line PQ is divided in the ratio of m : n.

We know that, by section formula, if a point (x, y) divides the line joining the points (x_{1}, y_{1}) and (x_{2}, y_{2}) in the ratio m : n, then the coordinates of intersection point (x, y) is given by:

,

Here, it is given that

P (2, – 3) = (x_{1}, y_{1})

Q (5, 6) = (x_{2}, y_{2}),

Let’s consider line is divided in the ratio of m : n

Hence the y-coordinate of point A:

y =

∴ 0 =

∴ – 3 m + 6 n = 0

∴ 3 m = 6 n

∴ m = 2 n

∴ m : n = 2 : 1

**Therefore, the line is divided in the ratio of 2 : 1.**

**(ii) Coordinates of intersection point:**

From the section formula, let’s find the value of x coordinate:

x =

∴ x =

∴ x = = 3

Since, value of y coordinate is 0 (being on X-axis)

**Therefore the coordinates of intersection point A are (3, 0).**

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