# How to give the sketch of a set

Mathematics Asked by rosita on October 6, 2020

I’m asked to give a sketch of this set: $$K = {(x,y)inmathbb R^2: 13x^2-10xy+13y^2=72}$$ and then give the points for which the distance from the origin is maximal/ will be maximal. Please help me. I have no idea how to solve it. Thanks in advance

Let $$x=rcos t, y=r sin t$$, so $$r^2(t)=frac{72}{13-10 sin t cos t}=frac{144}{26-10 sin 2t}.$$ So $$r^2(t)_{min}=r^2(t=3pi/4)=frac{144}{36},~r^2(t)_{max}=r^2(t=pi/4)=frac{144}{16}$$ $$r_{min}=2, r_{max}=3$$, these are the semi-minor and semi-major axes of an ellipse which are inclined at an angle $$3pi/4$$ and $$pi/4$$ with x-axis.

Answered by Z Ahmed on October 6, 2020

Analyze the given relation

• It is symmetric about $$x=y$$ as interchanging $$x$$ and $$y$$ yield the same equation
• It is symmetric about $$x+y=0$$ replacing $$y$$ by $$-x$$ and $$x$$ by $$-y$$ yield the same equation.
• It is a 2-degree equation in $$x$$ and $$y$$. So it must be a conic.
• It cannot be a circle beacuse the coefficient of $$xy$$ is non-zero
• It cannot be a parabola because a parabola has only one axis of symmetry whereas the curve already has two.
• The coefficients of $$x^2$$ and $$y^2$$ are of the same sign. So it cannot be a hyperbola as well.
• So the equation must represent an ellipse, with the axes being $$x=y$$ and $$x+y=0$$. So the centre is at $$(0,0)$$

So you need to find the maximum and minimum distances of a point on an ellipse from its centre. Evidently the points will be along the axes. So solve the given equation with $$x=y$$ and $$x+y=0$$ to get the points. One of them will represent the maximum distance and the other one will represent the minimum distance. The rest of the work is left to the reader.

Finally, this is how the ellipse looks like

The form of the equation should indicate to you that this is an ellipse with axes $$45^circ$$ rotated from the standard axes. As such, the LHS can be rewritten with $$(x+y)^2$$ and $$(x-y)^2$$: $$9(x-y)^2+4(x+y)^2=72$$ $$frac{(x-y)^2}8+frac{(x+y)^2}{18}=1$$ Therefore the nearest and farthest points from the origin correspond to vertices of the ellipse and have coordinates of the form $$(x,pm x)$$; you should find that the semi-major and semi-minor axes have lengths $$3$$ and $$2$$ respectively, with the semi-major axis parallel to $$x=y$$. From there you should be able to make a sketch.

Answered by Parcly Taxel on October 6, 2020

## Related Questions

### Is a surface curve made of planar points necessarily a line?

1  Asked on January 29, 2021 by mk7

### How to solve $sqrt{x!y!}=xy$ for $(x,y)inmathbb{Z}_{geq0}timesmathbb{Z}_{geq0}$?

3  Asked on January 29, 2021 by ramez-hindi

### Intuition for fractions of the localization of a non integral domain

1  Asked on January 29, 2021 by siddharth-bhat

### Proving origin to be removable singularity(Proof verification)

2  Asked on January 29, 2021

### Relationship between constants so that the center of curvature of the helix is contained in the cylinder

1  Asked on January 28, 2021

### An identity between integral

1  Asked on January 28, 2021 by inoc

### How to find the least in $E^{circ}=frac{5S^g}{162}+frac{C^circ}{50}+frac{2pi^2}{360}textrm{rad}$?

1  Asked on January 28, 2021 by chris-steinbeck-bell

### In what sense do we say two functions are equal?

0  Asked on January 28, 2021 by ziqi-fan

### $P (| X |> 1) = P (| X | <1)$

1  Asked on January 28, 2021

### Sequentially open sets but not open

1  Asked on January 28, 2021 by t-i

### Simplify $logleft(1+frac{x_i^2}{nu}right)$ with a $log(1+x)$ rule?

2  Asked on January 28, 2021

### Finding an Extremal for a function.

1  Asked on January 28, 2021 by zeroflank

### Isomorphism between group of homeomorphisms where $X nsim Y$

2  Asked on January 28, 2021

### Basis of the field $E$=$mathbb{Q}(sqrt{6}i-sqrt{5})$.

3  Asked on January 27, 2021 by questmath

### Limit points of the set ${frac {varphi(n) }n : nin mathbb{N}}$

1  Asked on January 27, 2021 by user-492177

### $forall epsilon >0,exists A in mathcal{A}$ such that $E subset A$ and $mu(A setminus E) < epsilon$

1  Asked on January 27, 2021 by user21

### Generalized Collatz: divide out by $2$’s and $3$’s, otherwise $5n+1?$

0  Asked on January 27, 2021 by rivers-mcforge

### How is the Rodrigues formula $L_n^k(x)=frac{e^x x^{-k}}{n!}frac{d^n}{dx^n}(e^{-x}x^{n+k})$ derived?

1  Asked on January 27, 2021 by almhz

### Absolute values of a closed set’s elements

2  Asked on January 27, 2021 by sicmath