The point on the curve x^2+2y=0 that is nearest the point (0, -1/2) occurs where y is?

Answer: 0

The point on the curve x^2+2y=0 that is nearest the point (0, -1/2) occurs where y is?

Answer: 0

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Baldwinsville, NY

Hi Sun,

Let's first rearrange to give an expression for y in terms of x:

x^{2} + 2y = 0

Subtracting x^{2} and dividing by 2:

y = (-x^{2})/2

If we plotted this, it would be a downward-opening parabola, centered on the y-axis.

Now, let's come up with an expression that would calculate the distance between a point on that parabola and any other point on the graph. The pythagorean theorem shows us how to calculate the length of the hypotenuse of a right triangle, and that's exactly what we would be dealing with here, if we treat the (difference in the y values) and (difference in x values) as the two perpendicular sides of a triangle.

So, we could say that the actual (hypotenuse) distance between two points would be √((y_{1}-y_{2})^{2} + (x_{1}-x_{2})^{2})

Let's treat x_{1} and y_{1} as the point along the parabola, and x_{2} and y_{2} as the (0, -1/2) point that is given. We want to determine the x_{1} and y_{1} points, so we will leave them as variables in our distance expression:

distance = √((y - (-1/2))^{2} + (x - 0)^{2})

Now, since we have an expression for y in terms of x, let's replace y with that expression, so that we will have only one variable in our equation:

distance = √((-x^{2}/2 + 1/2)^{2} + x^{2})

or

distance = ((-x^{2}/2 + 1/2)^{2} + x^{2})^{1/2}

Let's expand the first squared term, using FOIL:

distance = ((x^{4}/4 - x^{2}/2 + 1/4) + x^{2})^{1/2}

Combining the x^{2} terms:

distance = ((x^{4}/4 + x^{2}/2 + 1/4)^{1/2}

Knowing that we want to find the point where this distance is a minimum, just from examining the equation, the smallest we could ever make this distance would be (1/4)^{1/2} = 1/2, if x = 0. Because of the fact that all of the x exponents are even, any other x-value that we could possibly pick, positive or negative, would only make the distance larger.

So, we know that the distance is minimized when x=0. Since we want to know what the associated y-value is, we plug in 0 for x in the quadratic equation we derived from the given expression.

y = -(0)^{2}/2 = 0

This also checks with what we logically concluded to be the minimum possible distance: the distance between the points (0,0) and (0,-1/2) is 1/2.

Westford, MA

What is the y-value of the point on the curve x^2+2y=0 that is nearest the point (0, -1/2)?

Let’s find the distance between (0, -1/2) and a generalized point on the curve, (x,-x^2/2).

Draw a right triangle with horizontal and vertical legs. Label the end points of the hypotenuse (0, -1/2) and (x,-x^2/2). The distance, D, between the two points is the length of the hypotenuse, which we can get from the Pythagorean Theorem.

D^2 = (0-x)^2 + (-1/2 - -x^2/2)^2

D^2 = x^2 + (-1/2 + x^2/2)^2

D^2 = x^2 + ((x^2-1)/2)^2

D^2 = x^2 + ((x^2-1)^2)/4

4D^2 = 4x^2 + (x^2-1)^2

4D^2 = 4x^2 + x^4 - 2x^2 + 1

4D^2 = x^4 + 2x^2 + 1

4D^2 = (x^2 + 1)^2

D^2 = ((x^2 + 1)^2)/2^2

D = (x^2 + 1)/2

D = x^2/2 + 1/2, which is a parabola opening up with minimum when x = 0.

Substituting x = 0 into x^2+2y=0 => y = 0.

Let’s find the distance between (0, -1/2) and a generalized point on the curve, (x,-x^2/2).

Draw a right triangle with horizontal and vertical legs. Label the end points of the hypotenuse (0, -1/2) and (x,-x^2/2). The distance, D, between the two points is the length of the hypotenuse, which we can get from the Pythagorean Theorem.

D^2 = (0-x)^2 + (-1/2 - -x^2/2)^2

D^2 = x^2 + (-1/2 + x^2/2)^2

D^2 = x^2 + ((x^2-1)/2)^2

D^2 = x^2 + ((x^2-1)^2)/4

4D^2 = 4x^2 + (x^2-1)^2

4D^2 = 4x^2 + x^4 - 2x^2 + 1

4D^2 = x^4 + 2x^2 + 1

4D^2 = (x^2 + 1)^2

D^2 = ((x^2 + 1)^2)/2^2

D = (x^2 + 1)/2

D = x^2/2 + 1/2, which is a parabola opening up with minimum when x = 0.

Substituting x = 0 into x^2+2y=0 => y = 0.

Brookline, MA

Goal: Minimize D=√[x^{2}+(y-1/2)^{2}]

Constraint: G=x^{2}+2y=0

Let's minimize S=D^{2} instead. We will use Lagrange Multipliers.

S = x^{2}+y^{2}-y+1/4

We need ∇S=λ∇G

We get the system

(i) 2x = 2λx

(ii) 2y - 1 = 2λ

(iii) x^{2} + 2y = 0

Equation (i) is true if either x = 0 or λ = 1

If x = 0 then we have y = 0 for the point (0,0)

If λ = 1 then solving (ii) gives y = 1/2, and then there are no real solutions for x in (iii).

Hence the minimum is at (0,0) making y=0.

Daniel B.

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