Find the point on the line 7x+5y−2=0 which is closest to the point (−6,−2).
To find the point on the line 7x + 5y - 2 = 0 that is closest to the point (-6, -2), we need to find the perpendicular distance between the given point and the line.
First, let's rewrite the equation of the line in slope-intercept form, y = mx + b, where m represents the slope and b represents the y-intercept.
Rearranging the given equation, we have:
5y = -7x + 2
y = (-7/5)x + (2/5)
So the slope of the line is -7/5.
The slope of any line perpendicular to this line will be the negative reciprocal of -7/5, which is 5/7.
Now we can use the point-slope form of a line to find the equation of the line passing through (-6, -2) with a slope of 5/7.
Using the point-slope form:
y - y₁ = m(x - x₁)
y - (-2) = (5/7)(x - (-6))
y + 2 = (5/7)(x + 6)
Simplifying:
y + 2 = (5/7)x + 30/7
y = (5/7)x + 30/7 - 14/7
y = (5/7)x + 16/7
Now we have two lines:
1. The original line: y = (-7/5)x + (2/5)
2. The line passing through (-6, -2): y = (5/7)x + 16/7
To find the point where these two lines intersect, we can set them equal to each other and solve for x:
(-7/5)x + (2/5) = (5/7)x + (16/7)
Multiplying both sides of the equation by 35 to eliminate the denominators, we get:
-49x + 14 = 25x + 80
Adding 49x and subtracting 14 from both sides, we have:
-49x - 25x = 80 - 14
-74x = 66
Dividing both sides by -74, we find:
x = -66/74
Simplifying further, we have:
x = -33/37
Now, substitute this value of x back into either of the two line equations to find the value of y. Let's use the line passing through (-6, -2):
y = (5/7)(-33/37) + 16/7
y = -165/259 + 464/259
y = 299/259
So the point on the line 7x + 5y - 2 = 0 that is closest to the point (-6, -2) is approximately (-33/37, 299/259).
To find the point on the line 7x + 5y - 2 = 0 that is closest to the point (-6, -2), we can use the formula for the distance between a point and a line.
Step 1: Rewriting the line equation in slope-intercept form.
Rewrite the given equation 7x + 5y - 2 = 0 in slope-intercept form, which is y = mx + b.
To do this, isolate y by subtracting 7x from both sides of the equation:
5y = -7x + 2.
Now divide both sides of the equation by 5 to solve for y:
y = (-7/5)x + 2/5.
So, the slope-intercept form of the line is y = (-7/5)x + 2/5.
Step 2: Finding the slope of the line perpendicular to the given line.
To find the perpendicular slope, flip the sign of the slope of the given line and change the sign:
The given slope is -7/5, so the perpendicular slope is 5/7.
Step 3: Finding the equation of the line passing through the point (-6, -2) with the perpendicular slope.
Using the point-slope form of a line, where (x1, y1) is the given point and m is the perpendicular slope:
y - y1 = m(x - x1),
Substitute the given values (-6, -2) and the perpendicular slope (5/7) into the equation:
y - (-2) = (5/7)(x - (-6)).
Simplify the equation:
y + 2 = (5/7)(x + 6).
Step 4: Finding the point of intersection of the two lines.
To find the point of intersection, set the equations of the lines equal to each other:
(-7/5)x + 2/5 = (5/7)(x + 6).
Multiply both sides of the equation by 35 to get rid of the denominators:
-35(7/5)x + 35(2/5) = 35(5/7)(x + 6).
This simplifies to:
-49x + 14 = 25(x + 6).
Expand and simplify:
-49x + 14 = 25x + 150.
Rearrange the equation to isolate x:
-49x - 25x = 150 - 14.
Combine like terms:
-74x = 136.
Divide both sides of the equation by -74 to solve for x:
x = -136/74.
Simplify the fraction:
x = -68/37.
Step 5: Finding the y-coordinate of the point of intersection.
Substitute the value of x = -68/37 into the equation y = (-7/5)x + 2/5:
y = (-7/5)(-68/37) + 2/5.
Simplify the equation:
y = 476/185 + 2/5.
Find a common denominator and add the fractions:
y = (476/185) + (2/5) = (476/185) + (74/185) = 550/185.
Simplify the fraction:
y = 10/37.
Step 6: The point on the line closest to (-6, -2) is (-68/37, 10/37).
the shortest distance will be along the line perpendicular, which passes through (-6,-2)
The given line has slope -7/5, so the perpendicular has slope 5/7
y+2 = 5/7 (x+6)
The two lines intersect at (-33/37,61/37)
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Or, you can figure the distance to (-6,-2) knowing that y = (2-7x)/5:
d^2 = (-6-x)^2 + (-2-((2-7x)/5))^2)
= 1/25 (74x^2 + 132x + 1044)
dd/dx = (37x+33)/√(nonzero junk)
(x,y) = (-33/37,61/37)
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Or, you can find where the circle at (-6,-2) is tangent to the line. That will be where
(x+6)^2 + (y+2)^2 = r^2 and
y = (2-7x)/5
intersect in a single point.
25(x+6)^2 + (7x-2)^2 = 25r^2
74x^2 + 272x + 904-24r^2 = 0
If there is a single solution, the discriminant is zero, so
262^2 - 4(74)(904-24r^2) = 0
r^2 = 49375/1776
So, we have
25(x+6)^2 + (7x-2)^2 = 25*49375/1776
7x+5y−2=0
I trust we will come up with the same point. You can verify if you wish.