- #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.
## 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.