The following equations represent two ellipses:

$(x-2{)}^2+(y-3+2x{)}^2=5$

$2(x-3{)}^2+(\frac{y}{3}{)}^2=4.$

Write a MATLAB script 'ellipsesTask.m$\text{'}$ to

Task $1$: plot the two ellipses in one figure using Matlab function plot and;

Task $2$: find the coordinates of all the intersections $($there are $4$ of them$)$ of the two ellipses defined in

Eqs. $(1)$ and $(2).$

Key Requirements:

The solution should be contained in one MATLAB script named as 'ellipsesTask.m$\text{'}$

Task $1$

plot the two ellipses in one figure using MATLAB function plot only

The plot should be clear and contains correct labelling

Task $2$

Convert the intersection$-$finding problems into root$-$finding problems in the form of $f(x)=0$

where all four $f(x)$ need to be explicitly defined in the MATLAB code

Implement Newton's method and use it to find and print the coordinates $(x,y)$ of all the inter$-$

sections of the two ellipses. Note: you can use any online derivative finder to find $f^{\text{'}}(x)$

The Newton's method implemented should stop based on a given error margin, e$.$g$.=0\mathrm{.}001$

your MATLAB script shouldn't use any symbolic functions such as sym$,$ syms$.$

Submission:

A Matlab script.

$\%$ Task $1$: Plot the two ellipses in one figure using Matlab function plot

figure;

$\%$ Define the range for x and y

x$\_$range $=-10$:$0\mathrm{.}1$:$10$;

y$\_$range $=-10$:$0\mathrm{.}1$:$10$;

$\%$ Initialize matrices to store the points satisfying each ellipse equation

ellipse$1\_$points $=[]$;

ellipse$2\_$points $=[]$;

$\%$ Loop through the points and check if they satisfy the ellipse equations

for x $=$ x$\_$range

for y $=$ y$\_$range

if abs$(($x$-2)^2+($y$-3+2*$x$)^2-5)<mtext\; id="EditorElement\_60277"></mtext><mn\; id="EditorElement\_60278">0</mn><mn\; id="EditorElement\_60279">.</mn><mn\; id="EditorElement\_60280">1</mn>$

ellipse$1\_$points $=[$ellipse$1\_$points; x$,$ y$]$;

end

if abs$(2*($x$-3)^2+($y$/3)^2-4)<mtext\; id="EditorElement\_60390"></mtext><mn\; id="EditorElement\_60391">0</mn><mn\; id="EditorElement\_60392">.</mn><mn\; id="EditorElement\_60393">1</mn>$

ellipse$2\_$points $=[$ellipse$2\_$points; x$,$ y$]$;

end

end

end

$\%$ Plot the ellipses

plot$($ellipse$1\_$points$($:$,1),$ ellipse$1\_$points$($:$,2),\text{'}$b$-\text{'},$ 'LineWidth', $2)$;

hold on;

plot$($ellipse$2\_$points$($:$,1),$ ellipse$2\_$points$($:$,2),\text{'}$r$-\text{'},$ 'LineWidth', $2)$;

$\%$ Add labels and title

xlabel$(\text{'}$x$\text{'})$;

ylabel$(\text{'}$y$\text{'})$;

legend$(\text{'}$Ellipse $1\text{'},$ 'Ellipse $2\text{'})$;

title$(\text{'}$Two Ellipses'$)$;

grid on;

hold off;

$\%$ Task $2$: Find the coordinates of all the intersections $($there are $4$ of them$)$ of the two ellipses defined in Eqs. $(1)$ and $(2)$

syms x y;

$\%$ Define the ellipse equations

ellipse$1=($x $-2)^2+($y $-3+2*$x$)^2==5$;

ellipse$2=2*($x $-3)^2+($y$/3)^2==4$;

$\%$ Find the intersection points

$[$x$\_$intersect, y$\_$intersect$]=$ solve$($ellipse$1,$ ellipse$2,$ x$,$ y$)$;

$\%$ Convert the symbolic solutions to numeric values

x$\_$intersect $=$ double$($x$\_$intersect$)$;

y$\_$intersect $=$ double$($y$\_$intersect$)$;

$\%$ Display the intersection points

fprintf$(\text{'}$The intersection points are:

$\text{'})$;

for i $=1$:length$($x$\_$intersect$)$

fprintf$(\text{'}(\%\mathrm{.}4$f$,\%\mathrm{.}4$f$)$

$\text{'},$ x$\_$intersect$($i$),$ y$\_$intersect$($i$))$;

end

$($This code is giving me scattered ellipses. I want continous ellipses please. $)$