Optimization of the distance from the point on an ellipse

In summary, the conversation discusses the process of maximizing a function subject to a constraint, specifically solving for the maximum value of the function D with the constraint 4x^2 + 5y^2 = 20. The mistake made was not considering the bounds of the variable y, which resulted in an incorrect answer. The correct answer was found by checking the endpoints of y= -2 and y=2 and finding that the maximum value was obtained at y=2.
  • #1
Saptarshi Sarkar
99
13
Homework Statement
Find the point on an ellipse ##4x^2 + 5y^2 = 20## that is farthest away from the point (0,-2).
Relevant Equations
##D=\sqrt{x^2+(y+2)^2}##
My Attempt :We need to maximize

## D=\sqrt{x^2+(y+2)^2} ##

subject to the constraint

##4x^2 + 5y^2 = 20##.

From the constraint equation, we can write

##x^2=\frac{20-5y^2}{4}##

Using this in the formula for distance,

##D=\sqrt{\frac{20-5y^2}{4}+(y+2)^2}##

Differentiating this wrt y, and equating it to 0,

##\frac{5y}2=4y+8##

Solving this, we get y = 8

But, this can't be the answer as it doesn't even lie on the ellipse. The correct answer should be (0,2) but I didn't even get 2 as a possible answer.

What did I do wrong?

PS : I know that I can use Lagrange's method. I did that and got the correct answer. But I want to know what is the mistake that I made in this one.
 
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  • #2
You have to think about bounds: You are trying to maximize the function ##D(y)=\sqrt{\frac{20-5y^2}{4}+(y+2)^2}## on the interval ##y\in [-2,2]## (since this is the range of possible ##y## values). Since you found that ##D## has no critical points in the interior of this interval, the maximum value must be obtained at an endpoint- so you should check ##y=-2## and ##y=2## individually, and find that ##D## is maximized at ##y=2##.
 
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  • Informative
Likes Saptarshi Sarkar

1. What is the distance from a point to an ellipse?

The distance from a point to an ellipse is the shortest length between the point and any point on the ellipse.

2. How is the distance from a point to an ellipse calculated?

The distance from a point to an ellipse can be calculated using the distance formula, which takes into account the coordinates of the point and the center of the ellipse, as well as the length and width of the ellipse.

3. What is the optimal distance from a point to an ellipse?

The optimal distance from a point to an ellipse is the minimum distance, meaning the shortest possible distance between the point and any point on the ellipse.

4. How can the distance from a point to an ellipse be optimized?

The distance from a point to an ellipse can be optimized by finding the point on the ellipse that is closest to the given point, using mathematical techniques such as calculus or optimization algorithms.

5. Why is optimizing the distance from a point to an ellipse important?

Optimizing the distance from a point to an ellipse is important in various applications, such as navigation, robotics, and computer graphics. It allows for more accurate and efficient calculations and can improve the overall performance of a system.

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