Conic Sections Conic Sections 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.
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Generally the equation of hyperbola is given by: ⇒x2 / F2 - y2 / G2= 1; this is equation of hyperbola. 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.
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The general equation of a parabola is given by: - y2 = 4ax. equation of hyperbola is given by: - x2 / a2 – y2 / b2 = 1.
And the general
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. 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.
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