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Analyzing Conic Sections

TopCircle, ellipses, parabolas, and Hyperbola are all obtained when a cone is wedged by a plane. The general equation of any conic section is given by:
Ap2 + Bpq + Cq2 + Dp + Eq + f = 0;

If the value of B = 0 then we will see the ‘A’ and ‘C’ in the equations:

Name of conic section Relationship of A and C
Parabola A = 0 or C = 0 but both the values of ‘A’ and ‘C’ are never equals to 0.
Circle In case of Circle the value of ‘A’ and ‘C’ are both equal.
Ellipse In case of ellipse the sign of ‘A’ and ‘C’ are same but ‘A’ and ‘C’ are not equal.
Hyperbola In case of hyperbola the signs of ‘A’ and ‘C’ are opposite.

A line having curve shape is known as hyperbola. When the transverse axis of the given hyperbolas is aligned with the x – axis then the Equation of Hyperbola is given by:

$\frac{(p - h)^{2}}{a^{2}}$ - $\frac{(q-k)^{2}}{b^{2}}$ = 1

Having vertices - ( h $\pm$ a, k) and Foci (h $\pm$ c,k)

This is equation of hyperbola.

When the Intersection of Right Circular Cone and a plane which is also parallel to an element of the cone and the locus points of cone are equidistant from a fixed Point then we get a Plane Curve or a Parabola.

The general form is:

⇒ (αp + βq)2 + γp + δq + ∈= 0;

This above equation is obtained from the general conic sections equation which is given below:

⇒ Ap2+ Bpq + Cq2 + Dp + Eq + F = 0;

And the equations for a general form of parabola with the focus point F(s, t) and a directrix in the form:

⇒pa + qy + c = 0;

This is the equation of parabola.