The equation of line l is 3x-4y=24. the line intersects the x -axis at A and the y -axis at B. GIVEN THAT M is the point (4,-3) and O is the origin . find the length of AB .
To find the length of AB, we first need to find the coordinates of points A and B.
For point A, which is the x-intercept, we need to find the point where the line intersects the x-axis. To do this, we set y = 0 in the equation of the line:
3x - 4y = 24
3x - 4(0) = 24
3x = 24
x = 8
Therefore, point A is (8, 0).
For point B, which is the y-intercept, we need to find the point where the line intersects the y-axis. To do this, we set x = 0 in the equation of the line:
3(0) - 4y = 24
-4y = 24
y = -6
Therefore, point B is (0, -6).
Now that we have the coordinates of points A and B, we can calculate the length of AB using the distance formula:
AB = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Substituting the coordinates of A (8, 0) and B (0, -6) into the distance formula:
AB = sqrt((0 - 8)^2 + (-6 - 0)^2)
= sqrt((-8)^2 + (-6)^2)
= sqrt(64 + 36)
= sqrt(100)
= 10
Therefore, the length of AB is 10 units.
To find the length of AB, we need to determine the coordinates of points A and B first.
Point A is the point where line l intersects the x-axis. To find this point, we need to set y = 0 in the given equation:
3x - 4(0) = 24
Simplifying this equation, we get:
3x = 24
Dividing both sides by 3, we find:
x = 8
So, point A has coordinates (8, 0).
Similarly, point B is the point where line l intersects the y-axis. To find this point, we need to set x = 0 in the given equation:
3(0) - 4y = 24
Simplifying this equation, we get:
-4y = 24
Dividing both sides by -4, we find:
y = -6
So, point B has coordinates (0, -6).
Now that we have the coordinates of A and B, we can calculate the length of AB using the distance formula:
Distance AB = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Where (x1, y1) = (8, 0) and (x2, y2) = (0, -6)
Plugging in the values, we get:
Distance AB = sqrt((0 - 8)^2 + (-6 - 0)^2)
= sqrt((-8)^2 + (-6)^2)
= sqrt(64 + 36)
= sqrt(100)
= 10
Therefore, the length of AB is 10 units.