$\ln x$ function for drawing ellipses

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Hello dear mathematics enthusiasts,

last week, we saw general ellipse equation in the maths lesson.

Also an equation like $cx^{2}+dy^{2} = r^{2}$ is also usable, where equation forms like this.

Shortened Equation Declaration here.

Then I figured out that, $c$ is two times derivative of $-\ln a.$

Proof $c$ is two times derivative of $-\ln a$

What I want to understand is, how can we draw ellipses using linear functions and their derivatives.

Sorry for misunderstood, 4th-grade senior high school student.