Find the point on the line 4x+2y+5=0 which is closest to the point (-2,-1)
without calculus, the distance from the point to the line is
|4(-2)+2(-1)+5|/√(4^2+2^2) = √5/2
Using calculus, we have the distance z from (-2,-1) to (x,y) is
z^2 = (x+2)^2 + (y+1)^2
But y = -(4x+5)/2, so
z^2 = (x+2)^2 + ((-(4x+5)/2)+1)^2
z^2 = 5x^2 + 10x + 25/4
z = √(5x^2 + 10x + 25/4)
dz/dx = 10(x+1)/√(5x^2+10x+25/4)
dz/dx=0 at x=-1
z(-1) = √5/2
To find the point on the line 4x+2y+5=0 that is closest to the point (-2,-1), we can use the formula for the distance between a point and a line.
Step 1: Rearrange the given equation of the line into slope-intercept form (y = mx + b) to find the slope and y-intercept of the line.
4x + 2y + 5 = 0
2y = -4x - 5
y = -2x - 5/2
The slope of the line is -2.
Step 2: Use the formula for the distance between a point and a line to find the point on the line closest to (-2,-1). The formula is given by:
d = |Ax + By + C| / √(A^2 + B^2)
In this case:
A = 4
B = 2
C = 5
x = -2
y = -1
Substituting these values into the formula, we have:
d = |4(-2) + 2(-1) + 5| / √(4^2 + 2^2)
= |-8 - 2 + 5| / √(16 + 4)
= |-5| / √20
= 5 / 2√5
Step 3: To find the point on the line closest to (-2,-1), we need to find the point on the line with the same y-coordinate (-2x - 5/2).
Since the slope of the line is -2, the line is perpendicular to the given line. So the slope of the line perpendicular to it will be the negative reciprocal of -2, which is 1/2.
Using the point-slope form of a line, the equation of the line perpendicular to the given line and passing through (-2,-1) will be:
y + 1 = 1/2(x + 2)
y = 1/2x + 1
Step 4: Now we have two equations:
1) The equation of the given line: y = -2x - 5/2
2) The equation of the line perpendicular to it passing through (-2,-1): y = 1/2x + 1
By solving these two equations, we can find the coordinates of the point on the given line closest to (-2,-1).
-2x - 5/2 = 1/2x + 1
-2x - 1/2x = 1 + 5/2
-4x - x/2 = 1 + 5/2
-8x - x = 2 + 5
-9x = 7
x = -7/9
Substituting this value of x into any of the two equations:
y = -2(-7/9) - 5/2
y = 14/9 - 5/2
y = 28/18 - 45/18
y = -17/18
Step 5: Therefore, the point on the line 4x + 2y + 5 = 0 that is closest to the point (-2,-1) is (-7/9, -17/18).
To find the point on the line that is closest to the given point, you need to find the perpendicular distance between the line and the point. The point on the line that is closest to the given point will be the one where the perpendicular distance is minimized.
Step 1: Find the slope of the given line.
The given line is in the form Ax + By + C = 0, where A = 4, B = 2, and C = 5. The slope of the line can be found using the formula -A/B. So, the slope of the line is -4/2 = -2.
Step 2: Find the equation of the line perpendicular to the given line that passes through the given point.
The perpendicular slope to the line is the negative reciprocal of the slope of the line. So, the slope of the perpendicular line is 1/2 (the reciprocal of -2). Using the slope-intercept form (y = mx + b), we can substitute the values of the given point (-2, -1) to find the equation:
-1 = (1/2)(-2) + b
-1 = -1 + b
b = 0
Therefore, the equation of the line perpendicular to the given line that passes through the point (-2, -1) is y = (1/2)x.
Step 3: Find the intersection point of the two lines.
To find the intersection point of the two lines, we can solve the system of equations formed by the two lines' equations. So, we have the equations:
4x + 2y + 5 = 0 (given line)
y = (1/2)x (perpendicular line)
Substitute the value of y from the second equation into the first equation to solve for x:
4x + 2(1/2)x + 5 = 0
4x + x + 5 = 0
5x + 5 = 0
5x = -5
x = -1
Substitute the value of x back into the second equation to solve for y:
y = (1/2)(-1)
y = -1/2
Therefore, the intersection point of the two lines is (-1, -1/2).
Step 4: Calculate the distance between the intersection point and the given point.
Using the distance formula, we can find the distance between the given point (-2, -1) and the intersection point (-1, -1/2):
Distance = sqrt[(x₂ - x₁)^2 + (y₂ - y₁)^2]
Distance = sqrt[(-1 - (-2))^2 + (-1/2 - (-1))^2]
Distance = sqrt[(1)^2 + (-1/2 + 2/2)^2]
Distance = sqrt[1 + (1/2)^2]
Distance = sqrt[1 + 1/4]
Distance = sqrt[5/4]
Distance = sqrt(5)/2
Therefore, the point on the line 4x + 2y + 5 = 0 that is closest to the point (-2, -1) is the intersection point (-1, -1/2), and the distance between the two points is sqrt(5)/2.