how to find intersect with 2 lines

find the points of intersection between 𝑥^2+𝑦^2=3 and 𝑥𝑦=1. Convert this into a system of two nonlinear algebraic equations.solve a system of two nonlinear algebraic equations using multivariable Newton-Raphson method with an initial guess of 𝑥=1, 𝑦=1.

dx = 0.1;
x  = 0:0.1:5;
f2=xy-1
f1=x.^2+y.^2-3;

y2 = 1×51

1.7321 1.7349 1.7436 1.7578 1.7776 1.8028 1.8330 1.8682 1.9079 1.9519 2.0000 2.0518 2.1071 2.1656 2.2271 2.2913 2.3580 2.4269 2.4980 2.5710 2.6458 2.7221 2.8000 2.8792 2.9597 3.0414 3.1241 3.2078 3.2924 3.3779

figure(1)
p = plot(x, y1, x, y2);
p(1).LineStyle = "-.";
p(2).LineWidth = 2;

r                    = find(y1 == y2);
x_intersection       =  x(r);
x_value_intersection = y1(r);
figure(2)
p2=plot(x, y1, x, y2, x_intersection, x_value_intersection, 'o');

[Steven Lord]

This sounds like a homework assignment. If it is, show us the code you've written to try to solve the problem and ask a specific question about where you're having difficulty and we may be able to provide some guidance.

If you aren't sure where to start because you're not familiar with how to write MATLAB code, I suggest you start with the free MATLAB Onramp tutorial to quickly learn the essentials of MATLAB.

If you aren't sure where to start because you're not familiar with the mathematics you'll need to solve the problem, I recommend asking your professor and/or teaching assistant for help.

[Sam Chak]

Hi @JEEVITHA,

I understand that Newton–Raphson doesn't require plotting as long as it can find the intersection using the numerical method. However, as advised previously, you should plot the circle and the reciprocal function. Follow the steps provided, even though the homework question does not explicitly require you to plot them.

This will aid in your learning if you genuinely want to understand the material, not just for the sake of submitting the homework and passing the course.

[Sam Chak]

Hi @JEEVITHA

I see the reciprocal function in y1, but I don't see the circle in y2.

Can you please show the original question again?

Answer 1

[Sam Chak]

Hi @JEEVITHA

If you post like this, your question will likely be closed by the mods later. Please plot the circle and the reciprocal function on the same graph using the plot() function.

Do you know what the Newton–Raphson algorithm is? Without seeing the procedure, I cannot provide guidance. Of course, I can search on Google, but it's better if you search since this is your homework.

[Sam Chak]

Hi @JEEVITHA

The curves are incorrectly plotted because f1 and f2 in your code are unused. The circle requires some algebraic manipulations to plot upper and lower arcs.

As a hint, if you are proficient in pure math algebraic manipulations, you will arrive at the answers with the values related to the Golden Ratio. As you can see, the intersections are around x = ±0.6 and x = ±1.6.

% circle: x² + y² = 3
x1  = linspace(-sqrt(3), sqrt(3), 347);
y1a =   sqrt(3 - x1.^2);    % upper arc
y1b = - sqrt(3 - x1.^2);    % lower arc
plot(x1, y1a, x1, y1b, 'color', '#0920b8'), hold on

% reciprocal function: y = 1/x
x2a = linspace(0.35, 3, 266);
x2b = linspace(-3, -0.35, 266);
y2  = @(x) 1./x;
plot(x2a, y2(x2a), x2b, y2(x2b), 'color', '#b80909')

% labels
legend('Circle', '', 'Reciprocal fcn', '')
xlabel('x'), ylabel('y')
grid on
axis equal