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Calculate the minimum distance from the origin to the curve

$$3x^2+4xy+3y^2=20$$

The only method I know of is Lagrange multipliers. Is there any other method for questions of such type? Any help appreciated.

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8 Answers 8

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You can always try to write it in polar coordinates:

$$3r^2+4r^2\sin\theta\cos\theta=20$$ $$r^2=\frac{20}{3+2\sin2\theta}\ge\frac{20}{3+2}=4$$

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    $\begingroup$ Nice solution - excellent example of how change-of-coordinates can really help. $\endgroup$ Jan 9, 2017 at 15:47
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Hint: $4xy \le 2(x^2+y^2), 3x^2+3y^2 = 3(x^2+y^2) \implies 5(x^2+y^2) \ge 20 \implies x^2+y^2 \ge .....$

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  • $\begingroup$ sorry, i dont see how it helps. can u elaborate? $\endgroup$
    – Shobhit
    Jan 9, 2017 at 14:22
  • $\begingroup$ Maybe you coul add that the equality can be achieved and the bound is tight. $\endgroup$
    – user65203
    Jan 10, 2017 at 8:44
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This ellipse is centered at the orgin.

The distance you seek equals the length of the minor axis.

Next thing to know is that it has been rotate 45 degrees.

$x = \frac {\sqrt 2}{2} x' + \frac {\sqrt 2}{2} y'\\ y = -\frac {\sqrt 2}{2} x' + \frac {\sqrt 2}{2} y'$

Will rotate it into standard position.

Or, the egienvalues of $\begin{bmatrix} 3&2\\2&3\end{bmatrix}$ are $1$ and $5$

$x'^2 + 5y'^2 = 20\\ \frac {x'^2}{20} + \frac {y'^2}{4} = 1$

$2$

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By inspection (or see this for additional reference if required), we can see that this is an ellipse with centre at origin and axes of symmetry $y=\pm x$. Substituting into the ellipse equation gives end points of minor and major axes respectively: $$\\\begin{align} 3x^2\pm 4x^2+3x^2&=20\\ 2x^2, 10x^2, &=20\\ x&=\pm\sqrt{2}, \pm\sqrt{10}\end{align}$$ Distance from origin to semi-major/semi-minor axes is given by $\sqrt{x^2+y^2}=|x|\sqrt{2}$.
Shortest distance from origin to ellipse is semi-minor axis, i.e. $$\sqrt{2}\cdot \sqrt{2}=\color{red}2$$

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  • $\begingroup$ @RodrigodeAzevedo - Corrected - Thanks!. $\endgroup$ Jan 10, 2017 at 3:15
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We have the (non-convex) quadratically constrained quadratic program (QCQP)

$$\begin{array}{ll} \text{minimize} & \| \mathrm x \|_2^2\\ \text{subject to} & \mathrm x^{\top} \mathrm A \mathrm x = 1\end{array}$$

where

$$\mathrm A = \frac{1}{20} \begin{bmatrix} 3 & 2\\ 2 & 3\end{bmatrix}$$

Since $\mathrm A$ is symmetric, its eigenvalues are real. We define the Lagrangian

$$\mathcal L (\mathrm x, \mu) := \| \mathrm x \|_2^2 - \mu (\mathrm x^{\top} \mathrm A \mathrm x - 1)$$

Taking the partial derivatives and finding where they vanish, we obtain

$$(\mathrm I_2 - \mu \mathrm A) \, \mathrm x = 0_2 \qquad \qquad \qquad \mathrm x^{\top} \mathrm A \mathrm x = 1$$

Hence, the Lagrange multiplier is the inverse of the maximum eigenvalue of $\mathrm A$

$$\mu^* = \frac{1}{\lambda_{\max} (\mathrm A)} = 4$$

and the corresponding eigenspace is $\{ \gamma 1_2 \mid \gamma \in \mathbb R\}$. Let $\mathrm x^* = t 1_2$ be the minimizer. From $\mathrm x^{\top} \mathrm A \mathrm x = 1$,

$$t^2 = \frac{1}{1_2^{\top} \mathrm A 1_2} = 2$$

Therefore, $t = \pm \sqrt 2$ and $\mathrm x^* = \pm \sqrt 2 \, 1_2$. The distance between the ellipse and the origin is, thus, $$\| \mathrm x^* \|_2 = \sqrt 2 \, \| 1_2 \|_2 = 2$$

enter image description here


Related:

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  • $\begingroup$ Awesome approach _ $\endgroup$
    – Shobhit
    Jan 10, 2017 at 2:49
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Taking into account that it is an ellipse with short axis $y=x$ find crossings of this line with the ellipse.

enter image description here

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we have $$d(x,y)=\sqrt{x^2+y^2}$$ we get that $y$ from the curve: $$y_{1,2}=-\frac{2}{3}x\pm\sqrt{\frac{20}{3}-\frac{5}{9}x^2}$$ can you proceed?

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    $\begingroup$ substituting and differentiating to find minima would be very long method. $\endgroup$
    – Shobhit
    Jan 9, 2017 at 14:21
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    $\begingroup$ should i do it for you? $\endgroup$ Jan 9, 2017 at 14:23
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    $\begingroup$ oh no, i just want to know if there is a smarter, shorter method to solve above question. $\endgroup$
    – Shobhit
    Jan 9, 2017 at 14:24
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Let be $(x_0,y_0)$ a point of max/min distance the the origin. The (easily calculable) tangent to the ellipse in $(x_0,y_0)$ must be orthogonal to the line $x_0 y = y_0 x$.

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