How to solve the equation of this identity to obtain the specific values of coefficients A, B, and C？

% Clear all variables
clear all;
close all;
clc;

% Define symbolic variables
syms x y k m A B C x0 y0 a b ln;

% Define the line equation
line = y == k*x + m;

% Define the equation eq
eq = A*((y - y0)/(x - x0))^2 + B*((y - y0)/(x - x0)) + C == 0;

% Define the line l1
l1 = y - y0 == k*(x - x0) + m - y0 + k*x0;

% Define the line l2
l2 = (y - y0 - k*(x - x0))/(m - y0 + k*x0) == 1;

% Extract the left - hand side of l2
ln = lhs(l2);

% Define the left - hand side and right - hand side of the equation
leftSide = A*((y - y0)/(x - x0))^2 + B*((y - y0)/(x - x0)) + C;
rightSide = (((x - x0 + x0*ln)^2/a^2 + (y - y0 + y0*ln)^2/b^2 - (ln)^2)/(x - x0)^2);

% Solve the identity equation
sol = solve(leftSide == rightSide, [A, B, C], 'ReturnConditions', true);

% Display the result
disp('Result of solving the identity equation:');

Result of solving the identity equation:

disp(sol);

             A: -(z1*a^2*b^2*k^2*x^2*x0^2 + a^2*b^2*k^2*x^2 - 2*z1*a^2*b^2*k^2*x*x0^3 + z*a^2*b^2*k^2*x*x0^2*y - z*a^2*b^2*k^2*x*x0^2*y0 - 2*a^2*b^2*k^2*x*x0 + z1*a^2*b^2*k^2*x...
             B: z
             C: z1
    parameters: [z    z1]
    conditions: y ~= y0

% Solve for A, B, C that hold for all x, y
assume(x, 'real');
assume(y, 'real');
eq2 = leftSide == rightSide;
sol2 = solve(eq2, [A, B, C], 'ReturnConditions', true);

% Display the result
disp('Result of solving for A, B, C that hold for all x, y:');

Result of solving for A, B, C that hold for all x, y:

disp(sol2);

             A: -(z1*a^2*b^2*k^2*x^2*x0^2 + a^2*b^2*k^2*x^2 - 2*z1*a^2*b^2*k^2*x*x0^3 + z*a^2*b^2*k^2*x*x0^2*y - z*a^2*b^2*k^2*x*x0^2*y0 - 2*a^2*b^2*k^2*x*x0 + z1*a^2*b^2*k^2*x...
             B: z
             C: z1
    parameters: [z    z1]
    conditions: y ~= y0

The code is as above, but the result obtained after running is not what I want. A, B, and C can only be expressed in terms of other parameters. Why does the result include z? How can I get the specific parameter values?

[Torsten]

One equation for three unknowns ? That doesn't make much sense.

[Christopher]

Theoretically, it can be solved. Since it is an identity, the corresponding coefficients can be equated to find the solution.

[Torsten]

Which coefficients ? What about A = 0, B = 0, C = rightSide ?

[Sam Chak]

Hi @Christopher,

Could you provide a geometric sketch of the lines? This would help us intuitively understand what you aim to solve.

Answer 1

[Sameer]

Hi @Christopher

It looks like your symbolic solution is returning parameters 'z' and 'z1' because the system of equations is underdetermined, there are infinitely many solutions for 'A', 'B', and 'C' in terms of these parameters. To obtain specific values for 'A', 'B', and 'C', you need additional constraints or specific values for other variables.

Hope this helps!