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I am new to conic sections.
A tangent to any conic section is given by $$ Axx_1+Byy_1+h(x+x_1)(y+y_1)+g(x+x_1)+f(y+y_1)+c=0 $$ which applies to parabolas, circles, ellipses, hyperbolas. However the point of contact in slope form looks arbitrary.
For example: point of contact of tangent $\; y=mx+c\; $ to parabola $\; y^2=4ax\;$ is $\;\left( \frac{a}{m}, \frac{2a}{m}\right)\; $
and to parabola $\;x^2=4ay\;$ is $\;\left(2am, am^2\right).$
They seem to have no connection to me. Is there any way to establish a common formula for point of contact of tangents and conic sections as that of equation of tangents?

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That's the fundamental relation Pole-Polar line .

The pole is the point from where you are conducting the tangents to the conic, and the associated polar line is the line connecting the points of tangency.

To appreciate the most general and comprehensive results you shall work with homogeneous coordinates, in the real or better complex plane.

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