The end points of a line segment AB are A(a,b) and B(b,a), where a and b both are positive . In which ratio the line segment AB is divided by axes?
If both points are in the 1st quadrant, then the axes do not divide the line segment at all! AB does not intersect the axes.
Good
To find the ratio in which the line segment AB is divided by the axes, we need to find the coordinates of the point where the line segment intersects each axis.
Let's find the point of intersection with the x-axis:
For any point on the x-axis, the y-coordinate is always 0. So, at the point of intersection with the x-axis, we have y = 0.
Using the coordinates of point A (a, b), we can write the equation:
b = 0
This implies that b = 0.
Now, let's find the point of intersection with the y-axis:
For any point on the y-axis, the x-coordinate is always 0. So, at the point of intersection with the y-axis, we have x = 0.
Using the coordinates of point B (b, a), we can write the equation:
b = 0
This implies that a = 0.
Therefore, the line segment AB is divided by the axes in the ratio of a:0 or 0:b.
Since a and b are both positive, the ratio is 0:b.
To find the ratio in which line segment AB is divided by the axes, we need to determine the coordinates of the point where the line segment intersects each axis.
Let's start with the x-axis:
The x-axis has a y-coordinate of 0. To find the x-coordinate where the line segment AB intersects the x-axis, we substitute y = 0 into the equation of the line segment AB.
For point A(a, b):
0 = b - ax
ax = b
x = b/a
So, the line segment AB intersects the x-axis at the point (b/a, 0).
Now let's move on to the y-axis:
The y-axis has an x-coordinate of 0. To find the y-coordinate where the line segment AB intersects the y-axis, we substitute x = 0 into the equation of the line segment AB.
For point A(a, b):
y = b - 0/a
y = b
So, the line segment AB intersects the y-axis at the point (0, b).
Now, we can find the ratio in which the line segment AB is divided by the axes:
For the x-axis, the ratio is given by the distance of the point where AB intersects the x-axis from point A to the distance from point B to the x-axis.
This can be written as:
(b - 0)/(b - b/a) = b/(b - b/a)
For the y-axis, the ratio is given by the distance of the point where AB intersects the y-axis from point A to the distance from point B to the y-axis.
This can be written as:
(0 - a)/(0 - 0) = -a/0
However, division by zero is undefined, so the ratio for the y-axis is not applicable.
Therefore, the line segment AB is divided by the x-axis in the ratio of b/(b - b/a).