Calculus question

Question

 Find the point on the line −5x+7y−5=0−5x+7y−5=0 which is closest to the point (1,−2)(1,−2).

Answer is :



Answer 1

Finding the point on the line -5x + 7y - 5 = 0 that is closest to the point (1, -2) doesn't require calculus but can be done using calculus. You want to minimize the distance. To do so, we start with the distance equation:

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

We can plug in the point (1, -2) as one of the points. Let's choose (x₁, y₁) which then makes the equation:

d = √[(x₂ - 1)² + (y₂ + 2)²]

The other point (x₂, y₂) is on the line so must comply with the equation -5x + 7y - 5 = 0 or 7y = 5x + 5 or y = (5x)/7 + 5/7 so we can substitute "(5x)/7 + 5/7" for y₂ and just use "x" for x₂ giving us:

d = √[(x - 1)² + ((5x)/7 + 5/7 + 2)²] then simplify:

d = √[(x - 1)² + ((5x)/7 + 5/7 + 14/7)²]

d = √[(x - 1)² + ((5x)/7 + 19/7)²]

d = √[(x² - 2x + 1) + ((25x²)/49 + (190x)/49 + 361/49)]

d = √[x² + (25x²)/49 - 2x + (190x)/49 + 1 + 361/49]

d = √[(49x²)/49 + (25x²)/49 - (98x)/49 + (190x)/49 + 49/49 + 361/49]

d = √[(74x²)/49 + (92x)/49 + 410/49]

d(x) = (1/7)[74x² + 92x + 410]^1/2

Then take the derivative (using power rule and chain rule) and set it equal to zero and solve.

Note that the only way it can equal zero is when the numerator equals zero.

d'(x) =(1/14)[(74x² + 92x + 410]^-1/2 • [148x + 92]

1/14[(74x² + 92x + 410]^-1/2 • [148x + 92] = 0

148x + 92 = 0

x = -23/37

y = (5x)/7 + 5/7 = 5(-23/37)/7 + 5/7 = 10/37

So the point is (-23/37, 10/37)

HOWEVER, the easy way to solve this problem is using geometry. The point closest will be a point on the line that is perpendicular to -5x + 7y - 5 = 0 (which has a slope of 5/7) so the perpendicular has a slope of -7/5 and since it passes through (1, -2) the line is:

y + 2 = -7/5(x - 1)

To find the point on the line -5x + 7y - 5 = 0 that is also on y + 2 = -7/5(x - 1), solve the system of equations,

-5x + 7y - 5 = 0 is -5x + 7y = 5

y + 2 = -7/5(x - 1) is y = -7/5x - 3/5 or 7x + 5y = -3

-5x + 7y = 5 is -35x + 49y = 35

7x + 5y = -3 is 35x + 25y = -15

-35x + 49y = 35

35x + 25y = -15

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74y = 20

y = 20/74 = 10/37

Then using -5x + 7y = 5 or x = (7y)/5 - 1 so x = 7(10/37)/5 - 1 = -23/37