# How do you find the points on the parabola #2x = y^2# that are closest to the point (3,0)?

Points are (2.2) and (2, -2)

It is thus 3-x=1, x=2, and then y=2, -2. The nearest points are (2,2) and (2,-2)

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An alternative starts the same as bp's solution:

Our job now is to minimize the function:

(It should be clear that we can minimize the distance by minimizing the square of the distance.)

To minimize, differentiate, find and test critical numbers.

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To find the points on the parabola (2x = y^2) that are closest to the point ((3,0)), you can use the method of Lagrange multipliers. The distance between two points ((x, y)) and ((3, 0)) is given by the distance formula. By minimizing this distance, you can find the points on the parabola closest to ((3,0)).

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To find the points on the parabola (2x = y^2) that are closest to the point ((3, 0)), we need to minimize the distance between the given point and the points on the parabola.

The distance between two points ((x_1, y_1)) and ((x_2, y_2)) is given by the distance formula:

[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}]

For any point ((x, y)) on the parabola (2x = y^2), the distance (d) from ((3, 0)) can be expressed as:

[d = \sqrt{(x - 3)^2 + y^2}]

We want to minimize (d). Since (d) is a positive quantity, minimizing (d^2) is equivalent and easier. Thus, we minimize (d^2) instead:

[d^2 = (x - 3)^2 + y^2]

Substitute (2x = y^2) into the equation:

[d^2 = (x - 3)^2 + (2x)^2]

[d^2 = (x - 3)^2 + 4x^2]

Now, we take the derivative of (d^2) with respect to (x) and set it equal to zero to find the critical points:

[\frac{d}{dx}(d^2) = \frac{d}{dx}[(x - 3)^2 + 4x^2]]

[0 = 2(x - 3) + 8x]

[0 = 2x - 6 + 8x]

[0 = 10x - 6]

[x = \frac{6}{10} = \frac{3}{5}]

Substitute (x = \frac{3}{5}) into (2x = y^2) to find (y):

[2\left(\frac{3}{5}\right) = y^2]

[y^2 = \frac{6}{5}]

[y = \pm \sqrt{\frac{6}{5}}]

Thus, the points on the parabola (2x = y^2) that are closest to the point ((3, 0)) are (\left(\frac{3}{5}, \sqrt{\frac{6}{5}}\right)) and (\left(\frac{3}{5}, -\sqrt{\frac{6}{5}}\right)).

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