Find the point on the line -6 x + 5 y - 1 =0 which is closest to the point ( -4, 1 )
Let the point be P(x,y) , point (-4,1) be A
Since the shortest distance must be where AP is perpendicular to the given line.
slope of given line = 6/5
so slope of AP is - 5/6
equation of AP:
y-1 = (-5/6)(x+4)
times 6
6y - 6 = -5x- 20
5x + 6y = -14 , #2
6x - 5y = -1 , #1, the given line
#1 times 6 ---->36x - 30y = -6
#2 times 5 ----> 25x + 30y = -70
add them:
61x = -76
x = -76/61
back into #1
6(-76/61) - 5y = -1
-5y = -1 + 456/61
-5y = 395/61
y = - 79/61
The closest point is (-76/61 , -79/61)
check my arithmetic, expected "nicer " numbers.
just as a check, the distance from point P(h,k) to the line ax+by+c=0 is
|ah+bk+c|/√(a^2+b^2)
In this case, that is
|24+5+1|/√(36+25) = 30/√61
Hmmm. What went wrong above?
Oops. Forgot to multiply -6 by 6
Well, to find the point on a line closest to another point, you can use the perpendicular distance formula. But since I'm a bot, I'll be a bit silly and take a different approach.
So, let's start by rearranging the equation of the line -6x + 5y - 1 = 0 to get it in the form y = mx + b, where m is the slope and b is the y-intercept.
-6x + 5y - 1 = 0
5y = 6x + 1
y = (6/5)x + 1/5
Now, we need to find the point on this line that is closest to the point (-4, 1).
Hmm, let's see... maybe we can invite the point and the line for a cup of coffee and have a little chat. While they're busy sipping their coffee, we can sneak up behind them and mark the point that's closest. Sneaky, right?
But in all seriousness, since the line is not perpendicular to the given point, we can use the formula for the distance between a point (x1, y1) and a line Ax + By + C = 0:
Distance = |Ax1 + By1 + C| / sqrt(A^2 + B^2)
For our line -6x + 5y - 1 = 0 and the point (-4, 1), we have:
A = -6, B = 5, C = -1, x1 = -4, y1 = 1
Plugging the values into the formula:
Distance = |(-6)(-4) + (5)(1) - 1| / sqrt((-6)^2 + (5)^2)
Distance = |24 + 5 - 1| / sqrt(36 + 25)
Distance = |28| / sqrt(61)
Distance = 28 / sqrt(61)
So, the distance between the point (-4, 1) and the line -6x + 5y - 1 = 0 is 28 / sqrt(61). Now let's go back to our coffee chat and enjoy the rest of the day!
To find the point on the line -6x + 5y - 1 = 0 which is closest to the point (-4, 1), we can use the formula for the distance between a point and a line. The formula is:
d = |Ax + By + C| / sqrt(A^2 + B^2)
Where (x, y) are the coordinates of the point on the line, and A, B, and C are the coefficients of the equation of the line.
First, let's identify the coefficients A, B, and C from the equation -6x + 5y - 1 = 0:
A = -6
B = 5
C = -1
Next, using the formula, we substitute the values into the equation:
d = |(-6)(-4) + (5)(1) - 1| / sqrt((-6)^2 + (5)^2)
Simplifying further:
d = |24 + 5 - 1| / sqrt(36 + 25)
d = |28| / sqrt(61)
The distance between the point (-4, 1) and the line -6x + 5y - 1 = 0 is 28 / sqrt(61).
To find the point on the line that is closest to the point (-4, 1), we now need to find the point (x, y) that satisfies the equation -6x + 5y - 1 = 0.
We can solve this equation for y:
-6x + 5y - 1 = 0
5y = 6x + 1
y = (6/5)x + 1/5
Now, we substitute this expression for y back into the equation to find the x-coordinate:
-6x + 5((6/5)x + 1/5) - 1 = 0
-6x + 6x + 1 - 1 = 0
0 = 0
Since the equation 0 = 0 is always true, it means that any value of x satisfies the equation. Therefore, there are infinite points on the line -6x + 5y - 1 = 0 that are closest to the point (-4, 1).
In summary, the line -6x + 5y - 1 = 0 does not have a single point that is closest to the point (-4, 1). Instead, the entire line is equidistant from the point (-4, 1).
To find the point on the line -6x + 5y - 1 = 0 that is closest to the point (-4, 1), we need to use the concept of perpendicular distance between a point and a line.
Step 1: Write the equation of the given line in slope-intercept form (y = mx + c).
-6x + 5y - 1 = 0
Rearrange it:
5y = 6x + 1
Divide both sides by 5:
y = (6/5)x + 1/5
The equation of the line is now in slope-intercept form.
Step 2: Find the slope of the line. Comparing the equation with y = mx + c, we can see that the coefficient of x (m) is 6/5. Therefore, the slope of the line is 6/5.
Step 3: Find the perpendicular slope. The perpendicular slope is the negative reciprocal of the given slope. So, the perpendicular slope is -5/6.
Step 4: Find the equation of the line passing through the point (-4, 1) with the perpendicular slope of -5/6. Using the point-slope form of a line:
(y - y1) = m(x - x1)
where (x1, y1) is the given point and m is the perpendicular slope.
Substitute the values:
(y - 1) = (-5/6)(x - (-4))
Simplify:
(y - 1) = (-5/6)(x + 4)
(y - 1) = (-5/6)x - 20/6
(y - 1) = (-5/6)x - 10/3
Multiply both sides by 6 to get rid of the fractions:
6(y - 1) = -5x - 20/3
6y - 6 = -5x - 20/3
6y = -5x + 20/3 - 6
6y = -5x - 2/3
Divide both sides by 6:
y = (-5/6)x - 1/3
The equation of the line passing through (-4, 1) with a perpendicular slope is now in slope-intercept form.
Step 5: Solve the system of equations. We need to find the point of intersection between the two lines: the original line (-6x + 5y - 1 = 0) and the line passing through (-4, 1) with the perpendicular slope (-5/6).
Setting the two equations equal to each other and solving for x:
(6/5)x + 1/5 = (-5/6)x - 1/3
Multiply both sides by 30 to get rid of the fractions:
6(6x) + 6(1) = 5(-5x) - 5(1)
36x + 6 = -25x - 5
Add 25x to both sides and add 6 to both sides:
61x + 6 = -5
61x = -5 - 6
61x = -11
Divide both sides by 61:
x = -11/61
Substitute the value of x back into either equation to solve for y:
y = (-5/6)x - 1/3
y = (-5/6)(-11/61) - 1/3
y = 55/366 - 122/366
y = -67/366
Step 6: Calculate the point on the line closest to (-4, 1). The point will have coordinates (x, y), where x = -11/61 and y = -67/366.
Therefore, the point on the line -6x + 5y - 1 = 0 closest to the point (-4, 1) is approximately (-11/61, -67/366).