Find the point on the line 5x+5y+7=0
which is closest to the point (3,−4)
Just use your "closest distance from a given point to a line" formula
D = | 5(3) + 5(-4) + 7 |/√(5^2 + 5^2))
= 2/√50
= 2/5√2
= 2/5√2 units or √2/5 after rationalizing.
or , (the long and traditional way):
you know the slope of the given line is -1
so the slope of a perpendicular from (3,-4) is +1
equation of that perpendicular is:
y+4 = 1(x-3)
y = x - 7
sub into the given line to find the intersection:
5x + 5(x-7) + 7 0
5x + 5x - 35 + 7 = 0
10x = 28
x = 2.8
y = 2.8-7 = -4.2
distance between (3,-4) and (2.8, -4.2)
= √( (3-2.8)^2 + (-4+4.2)^2)
= √(.04+.04)
= √.08 = √(8/100)
= √8/10
= 2√2/10 = √2/5, same as above
To find the point on the line 5x + 5y + 7 = 0 which is closest to the point (3, -4), we can use the formula for the distance between a point and a line.
The formula for the distance between a point (x1, y1) and a line Ax + By + C = 0 is given by:
distance = |Ax1 + By1 + C| / √(A^2 + B^2)
In this case, A = 5, B = 5, C = 7, x1 = 3, and y1 = -4.
Plugging these values into the formula, we get:
distance = |5(3) + 5(-4) + 7| / √(5^2 + 5^2)
distance = |15 - 20 + 7| / √(25 + 25)
distance = |2| / √(50)
distance = 2 / √(50)
distance = 2 / (5√2)
distance = 2 / (5√2) * (√2 / √2)
distance = 2√2 / 10
distance = √2 / 5
So, the distance between the point (3, -4) and the line 5x + 5y + 7 = 0 is √2 / 5.
To find the point on the line closest to (3, -4), we need to find the point on the line that is perpendicular to the line connecting (3, -4) and the line 5x + 5y + 7 = 0.
The line 5x + 5y + 7 = 0 can be rewritten as:
y = (-5x - 7)/5
The slope of this line is -5/5 = -1.
The slope of the line perpendicular to this line is the negative reciprocal of -1, which is 1.
Using the point-slope form of a line, we can write the equation of the line perpendicular to 5x + 5y + 7 = 0 and passing through (3, -4) as:
y - (-4) = 1(x - 3)
y + 4 = x - 3
y = x - 7
Now we have two equations:
y = (-5x - 7)/5
y = x - 7
We can solve these equations simultaneously to find the point that satisfies both equations.
Setting the right-hand sides of the equations equal to each other, we have:
(-5x - 7)/5 = x - 7
Multiplying both sides of the equation by 5 to eliminate the fraction, we get:
-5x - 7 = 5x - 35
Adding 5x to both sides and adding 7 to both sides, we get:
-7 + 35 = 5x + 5x
28 = 10x
Dividing both sides of the equation by 10, we have:
x = 28/10
x = 2.8
Substituting this value of x into either of the two original equations, we find:
y = (-5(2.8) - 7)/5
y = (-14 - 7)/5
y = -21/5
So, the point on the line 5x + 5y + 7 = 0 which is closest to the point (3, -4) is approximately (2.8, -4.2).
To find the point on the line 5x + 5y + 7 = 0 which is closest to the point (3, -4), we can use the concept of perpendicular distance.
Step 1: Find the equation of the given line in slope-intercept form (y = mx + c).
5x + 5y + 7 = 0
5y = -5x - 7
y = (-5/5)x - 7/5
y = -x - 7/5
Step 2: The given line has a slope of -1. To find a line perpendicular to it, we need to take the negative reciprocal of the slope. So, the slope of the perpendicular line is 1.
Step 3: Use the equation of the perpendicular line and the given point (3, -4) to find the equation of the perpendicular line in point-slope form.
(y - y1) = m(x - x1)
(y - (-4)) = 1(x - 3)
y + 4 = x - 3
y = x - 7
Step 4: Solve the system of equations (the given line and the perpendicular line) to find the point of intersection. Set the equations equal to each other and solve for x and y.
-x - 7/5 = x - 7
2x = 7 - 7/5
2x = (35 - 7)/5
2x = 28/5
x = 14/5
Plugging the value of x back into one of the equations, we can find y:
y = x - 7
y = 14/5 - 7
y = -21/5
So, the point on the line 5x + 5y + 7 = 0 closest to the point (3, -4) is (14/5, -21/5).