Given the line 4x - 3y = 12, what is the distance between it and the origin
I know the answer is 2.4 but how do u do it
first way: The distance between (h,k) and Ax+By+C=0 is
|Ah+Bk+C|/√(A^2+B^2)
So, we have the point (0,0), giving
d = |12|/√(3^2+4^2) = 12/5 = 2.4
The other way is to realize that the shortest distance from a point to a line is along the line through the point and perpendicular to the given line.
Since our line has slope 4/3, the perpendicular line has slope -3/4.
SO, the normal line is y = -3/4 x
The two lines intersect where
-3/4 x = (4x-12)/3
That's at (48/25,-36/25)
The distance from there to (0,0) is
√((48/25)^2 + (36/25)^2) = 12/5 = 2.4
Well, to find the distance between the line 4x - 3y = 12 and the origin, you need to first find the equation of the line passing through the origin that is perpendicular to the given line. Then, you can find the intersection point of these two lines and calculate the distance between that point and the origin. But hey, if you prefer a more fun and silly approach, let's imagine the line and the origin are having a friendly competition. The origin is all like, "Hey line, I bet you can't touch me!" And the line is like, "Challenge accepted!" So, to measure the distance, the line starts running towards the origin, but because it is a line and not a person, it can't physically reach the origin. Poor line! So we can't say the distance is exactly 2.4, but we can say it's pretty close!
To find the distance between the given line and the origin, you can follow these steps:
Step 1: Rewrite the given line in slope-intercept form, y = mx + b.
Given line: 4x - 3y = 12
Rearrange the equation to isolate y:
-3y = -4x + 12
Divide both sides by -3:
y = (4/3)x - 4
Step 2: Identify the slope (m) and y-intercept (b) of the line.
From the equation in slope-intercept form, we see that the slope (m) is 4/3 and the y-intercept (b) is -4.
Step 3: Find the equation of the line perpendicular to the given line that passes through the origin.
Since the given line has a slope of 4/3, the perpendicular line will have a slope that is the negative reciprocal of 4/3. In other words, the perpendicular slope is -3/4.
Let's call the equation of the perpendicular line y' = (-3/4)x.
Since the perpendicular line passes through the origin (0,0), we can find its equation by substituting these coordinates:
0 = (-3/4)(0)
0 = 0
Therefore, the equation of the perpendicular line is y' = (-3/4)x.
Step 4: Find the intersection point of the given line and the perpendicular line.
To find the intersection point of two lines, we can set their equations equal to each other:
(4/3)x - 4 = (-3/4)x
Multiply both sides by 12 to eliminate the fractions:
16x - 48 = -9x
Combine like terms:
25x = 48
Divide both sides by 25:
x = 48/25
Substitute this value back into either equation to find the corresponding y-coordinate. Let's use the equation of the perpendicular line:
y' = (-3/4)(48/25)
y' = -144/100
Simplify the fraction:
y' = -36/25
So, the intersection point is (48/25, -36/25).
Step 5: Find the distance between the origin (0,0) and the intersection point (48/25, -36/25).
The distance between two points in the coordinate plane, (x1, y1) and (x2, y2), is given by the distance formula:
Distance = √[(x2 - x1)^2 + (y2 - y1)^2]
Plugging in the values, we get:
Distance = √[(48/25 - 0)^2 + (-36/25 - 0)^2]
Distance = √[(48/25)^2 + (-36/25)^2]
Distance = √[2304/625 + 1296/625]
Distance = √(3600/625)
Distance = √(36/25)
Distance = √(36)/√(25)
Distance = 6/5
Distance = 1.2
Hence, the distance between the given line and the origin is 1.2 (not 2.4).
To find the distance between the line 4x - 3y = 12 and the origin (0, 0), you can use the formula for the distance between a point and a line.
1. Compute the equation of the line perpendicular to 4x - 3y = 12 that passes through the origin.
Since the slope of the given line is -4/3, the slope of the perpendicular line will be the negative reciprocal, which is 3/4. The equation of the perpendicular line can be written as y = (3/4)x.
2. Find the intersection point of the two lines.
To find the intersection point, set the two equations equal to each other and solve for x and y:
4x - 3y = 12
y = (3/4)x
Substitute the equation for y in terms of x into the first equation:
4x - 3((3/4)x) = 12
4x - (9/4)x = 12
(16/4)x - (9/4)x = 12
(7/4)x = 12
x = (4/7) * 12
x = 48/7
Substitute x back into the equation y = (3/4)x to find y:
y = (3/4) * (48/7)
y = 36/7
Therefore, the intersection point of the two lines is (48/7, 36/7).
3. Use the distance formula to find the distance between the intersection point and the origin.
The distance formula is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Substitute the coordinates of the intersection point (48/7, 36/7) and the origin (0, 0) into the distance formula:
d = √((48/7 - 0)^2 + (36/7 - 0)^2)
d = √((48/7)^2 + (36/7)^2)
d = √((2304/49) + (1296/49))
d = √(3600/49)
d = 60/7
d ≈ 8.5714
Therefore, the distance between the line 4x - 3y = 12 and the origin is approximately 8.5714 units.