MHB Maximizing Area of an Ellipse Passing Through a Fixed Point

[leprofece]

347) An ellipse slmetric with respect to the coordinate axes handle: through the fixed point (h, k). Find The equation of the ellipse's?
area maximum
answer k<sup>2/SUP] h 2 + h 2 y 2 = 2 h 2 k2 The equations here must be
y = m(x-h)+k

Parametrizing
x= a cos(t)
y = b(sen(t)

D= (a cost-h)2+((macosth-h)2+ k -k)2

am I right?
Becuse I could not get the right answer</sup>

 

[Opalg]

347) An ellipse slmetric with respect to the coordinate axes handle: through the fixed point (h, k). Find The equation of the ellipse's?
area maximum
answer k<sup>2/SUP] h 2 + h 2 y 2 = 2 h 2 k2 The equations here must be
y = m(x-h)+k

Parametrizing
x= a cos(t)
y = b(sen(t)

D= (a cost-h)2+((macosth-h)2+ k -k)2

am I right?
Becuse I could not get the right answer</sup>

<sup>If I understand it correctly, the problem is to find an ellipse symmetric about the coordinate axes and passing through the fixed point $(h,k)$, such that the ellipse has the minimum possible area. (The question says maximum area, but in fact the only critical point is a minimum, and the maximum area is unbounded.)

The equation of the ellipse is $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, and we have to choose $a$ and $b$ so as to minimise the area of the ellipse, which is $\pi ab.$ The condition for the point $(h,k)$ to lie on the ellipse is $\dfrac{h^2}{a^2} + \dfrac{k^2}{b^2} = 1$, or $a^2k^2 + b^2h^2 = a^2b^2$. So we want to minimise $ab$ subject to the constraint $a^2k^2 + b^2h^2 = a^2b^2$. The usual way to solve problems of that sort is to use the method of Lagranga multipliers.</sup>

 

[leprofece]

If I understand it correctly, the problem is to find an ellipse symmetric about the coordinate axes and passing through the fixed point $(h,k)$, such that the ellipse has the minimum possible area. (The question says maximum area, but in fact the only critical point is a minimum, and the maximum area is unbounded.)

The equation of the ellipse is $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, and we have to choose $a$ and $b$ so as to minimise the area of the ellipse, which is $\pi ab.$ The condition for the point $(h,k)$ to lie on the ellipse is $\dfrac{h^2}{a^2} + \dfrac{k^2}{b^2} = 1$, or $a^2k^2 + b^2h^2 = a^2b^2$. So we want to minimise $ab$ subject to the constraint $a^2k^2 + b^2h^2 = a^2b^2$. The usual way to solve problems of that sort is to use the method of Lagranga multipliers.

thanks sorry but the students here haven't studied 2wo variables yet so it should be answerred or with one variable or with parametrics So I wrote parametrics in what I did
can you or anyone help me ??

 

[MarkFL]

Perhaps you could solve the constraint for either of the two variables and then use substitution to obtain an objective function in one variable...(Thinking)