Find the point on the line –2x+2y–3=0 which is closest to the point (–4–4).
Please provide solution. ( , )
Thanks guys!
quick way:
for a given point (p,q) , the shortest distance from that point to the line Ax + By + C = 0 is
|Ap + Bq + C|/)√(A^2+B^2)
so we have (p,q) = (-4,-4) and line
2x - 2y + 3 = 0
distance =|-8 +8 + 3|/(4+4)
= 3/√8
= 3/√8*√8/√8
= 3√8/8
= 6√2/8
= 3√2 /4
To find the point on the line that is closest to the given point, we need to find the perpendicular distance between the line and the point.
Step 1: Convert the given line equation into slope-intercept form (y = mx + c).
-2x + 2y - 3 = 0
2y = 2x + 3
y = x + 3/2
Step 2: Determine the slope (m) of the given line.
Comparing the equation to the slope-intercept form (y = mx + c), we can see that the slope (m) is 1.
Step 3: Determine the slope of the line perpendicular to the given line.
Since perpendicular lines have negative reciprocal slopes, the slope of the perpendicular line is -1.
Step 4: Write the equation of the perpendicular line in point-slope form using the slope (-1) and the given point (-4, -4).
Using the point-slope form (y - y1 = m(x - x1)), we have:
y - (-4) = -1(x - (-4))
y + 4 = -1(x + 4)
y + 4 = -x - 4
y = -x - 8
Step 5: Solve the system of equations formed by the two lines:
y = x + 3/2
y = -x - 8
By equating the expressions for y, we get:
x + 3/2 = -x - 8
Simplifying the equation, we have:
2x + 3 = -2x - 16
4x = -19
x = -19/4
Substituting this value back into one of the line equations (y = x + 3/2), we get:
y = (-19/4) + 3/2
y = -19/4 + 6/4
y = -13/4
Therefore, the point on the line -2x + 2y - 3 = 0 that is closest to the point (-4, -4) is (-19/4, -13/4).
So, the answer is ( -19/4, -13/4).
To find the point on the line –2x + 2y – 3 = 0 that is closest to the point (–4, –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:
d = |Ax1 + By1 + C| / sqrt(A^2 + B^2)
In this case, the line is –2x + 2y – 3 = 0, so A = -2, B = 2, and C = -3.
The point we want to find the closest point on the line to is (-4, -4), so x1 = -4 and y1 = -4.
Now we can substitute these values into the formula to calculate the distance:
d = |(-2)(-4) + (2)(-4) - 3| / sqrt((-2)^2 + (2)^2)
= |-8 - 8 - 3| / sqrt(4 + 4)
= |-19| / sqrt(8)
= 19 / sqrt(8)
This gives us the value of the distance between the point (-4, -4) and the line –2x + 2y – 3 = 0.
To find the coordinates of the point on the line that is closest to (-4, -4), we need to find the point on the line that has a perpendicular distance equal to the calculated distance.
The line –2x + 2y – 3 = 0 can be rewritten as 2y = 2x + 3, which simplifies to y = x + 3/2.
Since the given line has a slope of 1, the line perpendicular to it will have a slope of -1 (negative reciprocal). We can use this information to find the equation of the perpendicular line passing through (-4, -4).
The equation of a line with a slope of -1 passing through (-4, -4) can be written as:
y - y1 = m(x - x1), where m = -1 and (x1, y1) = (-4, -4).
Substituting in the values, we get:
y - (-4) = -1(x - (-4))
y + 4 = -x - 4
y = -x - 8
Now we need to solve for the intersection point of the lines y = x + 3/2 and y = -x - 8.
Setting the two equations equal to each other and solving for x:
x + 3/2 = -x - 8
2x + x = -8 - 3/2
3x = -16.5/2
x = -16.5/6
Substituting the value of x back into either of the equations, we can solve for y:
y = -(-16.5/6) - 8
y = 16.5/6 - 48/6
y = -31.5/6
Therefore, the point on the line –2x + 2y – 3 = 0 that is closest to the point (-4, -4) is approximately (-2.75, -5.25).
So the coordinates of the closest point are (-2.75, -5.25).
I want a line perpendicular to that line that passes through the point.
(There are quick formulas for this but I do not remember them)
the line in slope intercept form:
2 y = 2 x + 3
m = 2/2 = 1
so slope of out perpendicular is -1/1
= -1
so
y = - x + b
goes through (-4,-4)
-4 = 4 + b
so
y = -x -8
where does that hit our original line?
-2x +2(-x-8)) = 3
-4 x - 16 = 3
-4 x = 19
x = -19/4
y = -13/4