Find the point on the line -4x+7y+3=0 which is closest to the point(-5,-5)
easy way:
d = |(-4)(-5)+(7)(-5)+3|/√(4^2+7^2) = 12/√65
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calculus way:
the distance d can be found using
d^2 = (-5-x)^2 + (-5-y)^2
= (x+5)^2 + (5+(3+4x)/7)^2
d = 1/7 √(65x^2+794x+2669)
dd/dx = (130x+794) / 14√(65x^2+794x+2669)
dd/dx=0 when x = -373/65
so, y = -241/65
The distance between (-5,-5) and (-373/65,-241/65) = 12/√65
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Perpendicular line way:
perp line is
y+5 = -7/4 (x+5)
They intersect at (-373/65,-241/65)
and the distance is 12/√65
To find the point on the line -4x + 7y + 3 = 0 that is closest to the point (-5, -5), 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:
distance = |Ax₁ + By₁ + C| / √(A² + B²)
In this case, A = -4, B = 7, C = 3, and (x₁, y₁) = (-5, -5).
Plugging these values into the formula, we have:
distance = |-4(-5) + 7(-5) + 3| / √((-4)² + 7²)
distance = |20 - 35 + 3| / √(16 + 49)
distance = |-12| / √65
distance = 12 / √65
So, the distance between the point (-5, -5) and the line -4x + 7y + 3 = 0 is 12 / √65.
Now, to find the point on the line that is closest to (-5, -5), we need to find the point that lies on the line and has the same distance from (-5, -5) as the distance we just calculated.
We can use the formula for the distance between two points to calculate the distance between the point (x, y) on the line and (-5, -5):
distance = √((x - (-5))² + (y - (-5))²)
Since the distance is the same as 12 / √65, we can set the equation equal to this:
√((x - (-5))² + (y - (-5))²) = 12 / √65
Simplifying this equation and removing the square root, we have:
(x - (-5))² + (y - (-5))² = (12 / √65)²
(x + 5)² + (y + 5)² = (12 / √65)²
This equation represents a circle with center (-5, -5) and radius (12 / √65).
We can now solve this equation to find the coordinates of the point on the line that is closest to (-5, -5).
To find the point on the line -4x + 7y + 3 = 0 that is closest to the point (-5, -5), we can use the concept of the perpendicular distance between a point and a line.
1. First, let's write the equation of the line in slope-intercept form (y = mx + b):
-4x + 7y + 3 = 0
7y = 4x - 3
y = (4/7)x - 3/7
2. We know that the line perpendicular to the given line will have a slope that is the negative reciprocal of the slope of the given line. Therefore, the slope of the perpendicular line is -7/4 (opposite sign and flipped fraction).
3. Now we can find the equation of the line perpendicular to the given line and passing through the point (-5, -5). Using the point-slope form (y - y1 = m(x - x1)), we have:
y - (-5) = (-7/4)(x - (-5))
y + 5 = (-7/4)(x + 5)
y + 5 = (-7/4)x - 35/4
y = (-7/4)x - 55/4
4. Now we have two lines: the given line (-4x + 7y + 3 = 0) and the perpendicular line (-7/4x - 55/4). The point of intersection of these two lines will give us the point on the given line that is closest to (-5, -5).
5. To find the point of intersection, we set the two equations equal to each other and solve for x:
(-4x + 7y + 3) = (-7/4x - 55/4)
Multiply both sides by 4 to eliminate the fractions:
-16x + 28y + 12 = -7x - 55
Rearrange terms and simplify:
-16x + 7x + 28y = -55 - 12
-9x + 28y = -67
6. Solve the above equation for either x or y (let's solve for x to keep it consistent):
-9x = -67 - 28y
x = (67 + 28y)/9
7. Now substitute this value of x in either of the original equations (let's use the equation of the given line) to solve for y:
-4x + 7y + 3 = 0
-4((67 + 28y)/9) + 7y + 3 = 0
Simplify the equation to solve for y:
-268 - 112y + 63y + 27 = 0
-268 - 49y + 27 = 0
-241 - 49y = 0
-49y = 241
y = -241/49
8. Substitute the value of y back into the equation to solve for x:
x = (67 + 28(-241/49))/9
Simplify the equation to get the value of x:
x = (-1657/49)/9
x = -1657/441
9. Therefore, the point on the line -4x + 7y + 3 = 0 that is closest to the point (-5, -5) is approximately (-1657/441, -241/49).