The line segment joining (2, -3) and (5, 6) is divided by x-axis in the ratio:
a. 2:1
b. 3:1
c. 1:2
d. 1:3

Hint: We know that if any line segment joining to points let say A and B are divided by any other point C in a certain amount of ratio. Then the coordinates of C are given by the formula namely section formula, which is, \[{\text{(x,y) = (}}\dfrac{{{\text{mc + na}}}}{{{\text{m + n}}}}{\text{,}}\dfrac{{{\text{md + nb}}}}{{{\text{m + n}}}}{\text{)}}\], where (x,y) is coordinates of C, (a,b) and (c,d) are the coordinates of A and B.
Using section formula, on simplifying we’ll get the equation to simplify for the ratio in which the x-axis divides the line segment, so the coordinate of any point on the x-axis is (a,0). Using this we can find the required ratio.


Complete step by step Answer:

Given: x-axis divides the line segment joining (2, -3) and (5, 6)
Let the point where this line segment divides the x-axis be, (a, 0), and also let the x-axis divide this line segment in the ratio m: n as all the ordinate of the points in the x-axis is zero.
Now we know, if a line joining \[{\text{(a,b)}}\]and \[{\text{(c,d)}}\] gets divided by a point \[{\text{(x,y)}}\] in the ratio \[{\text{m:n}}\],
Then using section formula we can find the coordinates(x,y)
i.e. \[{\text{(x,y) = (}}\dfrac{{{\text{mc + na}}}}{{{\text{m + n}}}}{\text{,}}\dfrac{{{\text{md + nb}}}}{{{\text{m + n}}}}{\text{)}}\]
now according to the given data, we have the points as\[{\text{(a,b) = (2, - 3)}}\] , and \[{\text{(c,d) = (5,6)}}\]
So on substituting the above values we get,
\[{\text{(a,0) = (}}\dfrac{{{\text{m}} \times {\text{5 + n}} \times {\text{2}}}}{{{\text{m + n}}}}{\text{,}}\dfrac{{{\text{m6 + n( - 3)}}}}{{{\text{m + n}}}}{\text{)}}\]
Now on equating the y components, we get,
\[\dfrac{{{\text{m(6) + n( - 3)}}}}{{{\text{m + n}}}} = 0\]
Multiplying both sides by (m+n)
\[ \Rightarrow {\text{6m - 3n = 0}}\]
\[ \Rightarrow {\text{6m = 3n}}\]
Dividing both sides by 3
\[ \Rightarrow 2{\text{m = n}}\]
Now to get the value of ratio dividing both sides by n
\[\therefore \dfrac{{\text{m}}}{{\text{n}}}{\text{ = }}\dfrac{{\text{1}}}{{\text{2}}}\]
Hence the correct option is C.

Note: We use section formula to find the coordinates of a point which divides the line joining two points in a ratio, internally or externally. If the value of the ratio is negative then the point is external, whereas if the value of the ratio is positive then the point is internal.