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

To find the ratio in which the line segment is divided by the x-axis, we need to determine the distance of the point of intersection with the x-axis from the first point (2,-3) and the distance of the point of intersection with the x-axis from the second point (5,6).

Step 1: Find the equation of the line passing through the points (2,-3) and (5,6). The equation of a line passing through two points (x1, y1) and (x2, y2) is given by: (y - y1) = ((y2 - y1) / (x2 - x1)) * (x - x1)

Using the given points (2,-3) and (5,6), we can substitute the values into the equation to find the equation of the line: (y - (-3)) = ((6 - (-3)) / (5 - 2)) * (x - 2) (y + 3) = (9 / 3) * (x - 2) (y + 3) = 3 * (x - 2) y + 3 = 3x - 6 y = 3x - 6 - 3 y = 3x - 9

Step 2: Find the point of intersection with the x-axis. To find the point of intersection with the x-axis, we set y = 0 in the equation of the line: 0 = 3x - 9 3x = 9 x = 3

Step 3: Calculate the distances from the points (2,-3) and (5,6) to the point of intersection with the x-axis. Distance from (2,-3) to (3,0): d1 = sqrt((3 - 2)^2 + (0 - (-3))^2) = sqrt(1^2 + 3^2) = sqrt(1 + 9) = sqrt(10)

Distance from (5,6) to (3,0): d2 = sqrt((3 - 5)^2 + (0 - 6)^2) = sqrt((-2)^2 + (-6)^2) = sqrt(4 + 36) = sqrt(40) = 2 * sqrt(10)

Step 4: Calculate the ratio. The ratio is given by the distances d1 and d2: Ratio = d1 : d2 = sqrt(10) : 2 * sqrt(10) = 1 : 2

Therefore, the ratio in which the line segment joining (2,-3) and (5,6) is divided by the x-axis is 1:2.