# Conic Section - Cartesian Coordinates - Modified Form

Modified Form

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For some practical applications, it is important to re-arrange the standard form so that the focal-point can be placed at the origin. The mathematical formulation for a general conic section is then given in the polar form by

and in the Cartesian form by

begin{alignat}{7} sqrt{x^{2}+y^{2}} = left(l+e xright) \ Rightarrowleft(frac{x-frac{le}{1-e^{2}}}{frac{l}{1-e^{2}}}right)^{2}+frac{left(1-e^{2}right)y^{2}}{l^{2}} = 1 end{alignat}

From the above equation, the linear eccentricity (c) is given by .

From the general equations given above, different conic sections can be represented as shown below:

• Circle:
• Ellipse:
• Parabola:
• Hyperbola: