Finding the point on a parabola closest to a given point

Essentially, the coordinates nearest to P are x = 1 and y = 1, which can be found by substituting the given values into the equations and solving for y. In summary, the curve coordinates nearest to P are (1,1).
  • #1
leprofece
241
0
Find the curve coordinates nearest to P
a)x2 = y P(3,0)

ok to the boox exercise model I must do so

(x-3)2+ (y)2 = D

then x2-6x +9+y2 = D

introducing y into x
y-6sqrt(y) +9 + (y)2 = D

derivating
1-3/(sqrt(y) +0+2y

mcm = sqrt(y) -3 +2y = 0
solving
sqrt(y) = -2y+3
Squaring
y = 4y2-12y +9
4y2-13y +9 = 0
x = 1
y = 1
 
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  • #2
leprofece said:
Find the curve coordinates nearest to P
a)x2 = y P(3,0)

ok to the boox exercise model I must do so

(x-3)2+ (y)2 = D

then x2-6x +9+y2 = D

introducing y into x
y-6sqrt(y) +9 + (y)2 = D

derivating
1-3/(sqrt(y) +0+2y

mcm = sqrt(y) -3 +2y = 0
solving
sqrt(y) = -2y+3
Squaring
y = 4y2-12y +9
4y2-13y +9 = 0
x = 1
y = 1

Looks like you already figured this one out. :)
 

FAQ: Finding the point on a parabola closest to a given point

What is a parabola?

A parabola is a type of curve that is created by graphing a quadratic equation. It is a symmetrical curve that forms a "U" shape and can be found in many natural and mathematical phenomena.

How do you find the point on a parabola closest to a given point?

To find the point on a parabola closest to a given point, you need to use the distance formula to calculate the distance between the given point and any point on the parabola. Then, you can use calculus to find the minimum distance and corresponding point on the parabola.

What is the distance formula?

The distance formula is a mathematical formula that calculates the distance between two points, given their coordinates. It is represented as: d = √[(x2 - x1)^2 + (y2 - y1)^2], where (x1, y1) and (x2, y2) are the coordinates of the two points.

What is calculus and how is it used to find the point on a parabola closest to a given point?

Calculus is a branch of mathematics that deals with the study of change and motion. It is used to find the minimum and maximum points of a function, which is necessary for finding the point on a parabola closest to a given point. By finding the derivative of the distance formula and setting it equal to zero, we can solve for the x-coordinate of the minimum point, which can then be substituted into the original equation to find the corresponding y-coordinate.

Are there any other methods for finding the point on a parabola closest to a given point?

Yes, there are other methods such as using the vertex form of a parabola, which involves completing the square to find the vertex and then using the distance formula to calculate the distance between the vertex and the given point. However, the calculus method is the most efficient and accurate method for finding the closest point on a parabola.

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