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\le​ft(x\:+\:k​\right)^{\​frac{1}{3}​} y = a ( x + k ) 1 3 are orthogonal trajectories.

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Mauricio Alvarez
Mauricio Alvarez on 14 Sep 2020
Commented: Walter Roberson on 22 Apr 2021
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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Answers (1)

Pranav Verma
Pranav Verma on 17 Sep 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.
Please refer to syms documentation also to use diff and solve.
Also refer to the below discussions on the same lines:
Walter Roberson
Walter Roberson on 22 Apr 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.
The essential MATLAB functions to use are diff() and solve() ... just like @Pranav Verma said.
... 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 ?

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