the y=a(x+k)^1/3 thats the second y equation thats incorrect on the top
Find the value of the number LaTeX: a a such that the families of curves LaTeX: y=\left(x+c\right)^{-1} y = ( x + c ) − 1 and LaTeX: y\:=\:a\left(x\:+\:k\right)^{\frac{1}{3}} y = a ( x + k ) 1 3 are orthogonal trajectories.
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Mauricio Alvarez
el 14 de Sept. de 2020
Editada: Carlos Guerrero García
el 19 de Nov. de 2022
Find the value of the number a such that the families of curves y=(x+c)−1and y=a(x+k)13 are orthogonal trajectories.
im trying to write htis on matlab to have it solved can you help me
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Walter Roberson
el 14 de Sept. de 2020
Find the value of the number a such that families of curves and are othogonal trajectories
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Pranav Verma
el 17 de Sept. de 2020
Hi Mauricio,
As per my understanding, you intend to solve the above mentioned system of equations pertaining to the specified condition of trajectories being orthogonal. I'd suggest using the diff and solve function in MATLAB for achieving the above task.
Also refer to the below discussions on the same lines:
Thanks
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Walter Roberson
el 22 de Abr. de 2021
Orthoganal implies that the angle between angle between the two is 90 degrees. The angle of the original item is arctan of the tangent. Tangent is derivative. Derivative in MATLAB is diff() . So you have to start with diff() of the function, arctan, add pi/2, tan .
If you have calculated the derivative as y/x then tan(atan(y/x) + pi/2) works out as -x/y so you can skip the tan and arctan and go directly to taking the diff and calculating -1/diff as the angle of the orthoganal.
Now you equate the -1/diff and the second function and solve() for the parameter.
... Unless you want to count multiplying by negative 1 or division as an essential MATLAB function that people need to call out in their volunteer explanations ?
Carlos Guerrero García
el 16 de Nov. de 2022
Editada: Carlos Guerrero García
el 19 de Nov. de 2022
I know that the following code is only a visual interpretation, buit I think it would be nice for a visual approach to the problem
[x,y]=meshgrid(-5:0.1:5); % An adecuate region for the visualization
for a=-3:0.01:3;
contour(x,y,x-1./y,20,'r'); % Plotting the curves of the first family
hold on; % For do not clear the figure
contour(x,y,(y/a).^3-x,[-10:10],'b'); % Plotting the curves of the second family for each value of the parameter
axis([-5 5 -5 5]); axis equal; % For the same scale
title(['a= ',num2str(a)]);
drawnow;
hold off;
end
So, while you're visualizing the animation, try to find the answer to the following cuestion: WHEN the blue lines are orthogonal to the red ones ?
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