Question

The X-axis divides the line joining the points A(2,-3) and B(5,6) in the ratio of

• A. 1 : 2
• B. 2 : 1
• C. 3 : 5
• D. 2 : 3

Answer 1

The Correct Option is A

Given points:

\( A(2,-3) \) and \( B(5,6) \) 

Step 1: Use the section formula

The section formula states that if a point \( P(x,y) \) divides the line joining \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \), then:

\[ x = \frac{m x_2 + n x_1}{m+n}, \quad y = \frac{m y_2 + n y_1}{m+n} \]

Step 2: Apply the condition

The X-axis divides the line, so the y-coordinate of the dividing point is 0.

\[ \frac{m(6) + n(-3)}{m+n} = 0 \]

Step 3: Solve for \( m:n \)

\[ 6m - 3n = 0 \]

\[ 6m = 3n \]

\[ \frac{m}{n} = \frac{3}{6} = \frac{1}{2} \]

Final Answer: 1:2

Answer 2

To find the ratio in which the X-axis divides the line segment joining points A(2, -3) and B(5, 6), we use the section formula in coordinate geometry. 
According to the section formula, the coordinates of a point that divides a line segment joining A(x₁, y₁) and B(x₂, y₂) in the ratio m:n are:

( (mx₂ + nx₁) / (m + n), (my₂ + ny₁) / (m + n) )

 

Here, the point lies on the X-axis, which means its y-coordinate is 0. Therefore, we have:

(my₂ + ny₁) / (m + n) = 0

 

Substituting the coordinates of points A and B:

m(6) + n(-3) = 0

 

6m - 3n = 0

 

Rearranging gives:

6m = 3n

 

Dividing both sides by 3:

2m = n

 

Thus, the ratio m:n is 1:2. 
Therefore, the X-axis divides the line segment joining the points A(2, -3) and B(5, 6) in the ratio 1:2.