Conic Section Conic Section Conic section can be defined as a curve which is made by the intersection of a cone that resides on a plane. And in other terms it can be assumed as the plane algebraic curve with the degree of two.The general equation of any conic section is given by: Sp2 + Tpq + Uq2 + Vp + Wq + X = 0; If the value of T = 0 then we will see the ‘S’ and ‘U’ in the equations. Name of conic section Relationship of A and C. Parabola S = 0 or U = 0 but both the values of ‘S’ and ‘U’ are never equals to 0. Circle - In case of circle the value of ‘S’ and ‘U’ are both equal. Ellipse - In case of ellipse the sign of ‘S’ and ‘U’ are same but ‘S’ and ‘U’ are not equal. Hyperbola - In case of hyperbola the signs of ‘S’ and ‘U’ are opposite. Let's have small introduction about all conic sections. Hyperbola can be defined as a line in a graph that has curve shape. Generally the equation of hyperbola is given by: ⇒x2 / F2 - y2 / G2= 1; this is equation of hyperbola.
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In geometry a parabola is a special type of curve that has its own shape just like an arc and the point situated on a parabola is always equidistance from the locus and the directrix. The general form of parabola is given as: ⇒(αs + βt)2 + γs + δt + ∈= 0; This equation is obtained from the general conic sections equation. The equation is given by: Sp2+ Tpq + Uq2 + Vp + Wq + X = 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. We can determine the conic section into three types: The three types of conic section are: 1. Parabola – Parabola has eccentricity is equal to 1. 2. Circle and Ellipse – Its eccentricity is larger than 0 smaller than 1. 3. Hyperbola – It has eccentricity value greater than 1. These all are three types of conic sections and its eccentricity. There are some special categories defined for a circle that is – in some of the cases it is assumed as fourth type conic section and in some cases it is assumed as a special type of ellipse. If we intersect a plane and a circular then we get conic section. Some equation of conic section is defined which are shown below: The general equation of a circle is given by: - x2 + y2 = a2. The equation of an ellipse is given by: - x2 / a2 + y2 / b2 = 1. The general equation of a parabola is given by: - y2 = 4ax.
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And the general equation of hyperbola is given by: - x2 / a2 – y2 / b2 = 1. We can write all of the above equations in standard form and in parametric form. The main and important area of conic section where it is used, they are: - astronomy and projective geometry.
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