Question

chapter :section formula
1. In what ratio is the line segment joining X (2, -3) and Y (5, 6) divides by the x-axis?
Also, find the coordinates of the point of division.
 Hint: X axis so P(x,y) = (x,0). Use section formula, Keep the ratio as m and n

Answer 1

Answer :

In ratio 1:2 the X-axis will divide the line joining the given points .

The coordinates of the point of division is (3 , 0)

Given :

• The points are X(2 , -3) and Y(5 , 6)
• X-axis divides the given points

To Find :

• The ratio in which the X-axis divides the given point
• The coordinates of the point on X-axis

Formula to be used :

If (x , y) divides a line joining the points (x₁ , y₁) and (x₂ , y₂) in ratio of m:n , then x and y are given as

$\sf \star \: \: x = \dfrac{mx_{2} + nx_{1}}{m+n} \: \: , \: \: y = \dfrac{my_{2}+ ny_{1}}{m+n}$

Solition :

Let us consider the ratio be m:n and the point on X-axis be P(x , 0)

Using section formula :

For X- coordinate

$\sf \implies x = \dfrac{m\times 5 + n\times 2 }{m+n} \\\\ \sf x = \dfrac{5m + 2n}{m+n}$

Now for Y-coordinate

$\sf \implies 0 = \dfrac{m\times 6 + n\times (-3)}{m+n} \\\\ \sf \implies 0 = 6m - 3n \\\\ \sf \implies 3n = 6m \\\\ \sf \implies \dfrac{m}{n}=\dfrac{3}{6} \\\\ \sf \implies \dfrac{m}{n}=\dfrac{1}{2} \\\\ \sf \implies m:n = 1:2$

Thus the ratio is 1:2

Now putting the value the m:n to find the X-coordinates :

$\sf \implies x = \dfrac{5\times 1 + 2\times 2}{1+2} \\\\ \sf \implies x = \dfrac{5+4}{3} \\\\ \sf \implies x = \dfrac{9}{3} \\\\ \sf \implies x = 3$

Thus the point on X-axis which divides the points X(2 , -3) and Y(5 , 6) is (3 , 0)