Can you explain how do you generate the conic sections?

Can you explain how do you generate the conic sections?

Conic sections can be generated by intersecting a plane with a cone. A cone has two identically shaped parts called nappes. One nappe is what most people mean by “cone,” and has the shape of a party hat. Conic sections are generated by the intersection of a plane with a cone.

What are the 4 conic sections and their equations?

STANDARD FORMS OF EQUATIONS OF CONIC SECTIONS:

Circle (x−h)2+(y−k)2=r2
Hyperbola with horizontal transverse axis (x−h)2a2−(y−k)2b2=1
Hyperbola with vertical transverse axis (y−k)2a2−(x−h)2b2=1
Parabola with horizontal axis (y−k)2=4p(x−h) , p≠0
Parabola with vertical axis (x−h)2=4p(y−k) , p≠0

How do you identify an ellipse?

Ellipse: When x and y are both squared and the coefficients are positive but different. The equation 3×2 – 9x + 2y2 + 10y – 6 = 0 is one example of an ellipse. The coefficients of x2 and y2 are different, but both are positive.

How do you tell the difference between a circle and an ellipse equation?

The only difference between the circle and the ellipse is that in an ellipse, there are two radius measures, one horizontally along the x-axis, the other vertically along the y-axis. Clearly, for a circle both these have the same value.

How do you determine if an equation is a degenerate conic?

When the plane does intersect the vertex of the cone, the resulting conic is called a degenerate conic. Degenerate conics include a point, a line, and two intersecting lines. The equation of every conic can be written in the following form: Ax2 + Bxy + Cy2 + Dx + Ey + F = 0.

How many generators are in a cone?

In figure a below, we have a cone and a cutting plane which is parallel to one and only one generator of the cone. This conic is a parabola. If the cutting plane is parallel to two generators, this intersects nappes of the cone, and a hyperbola is obtained.

What is ellipse equation?

When the centre of the ellipse is at the origin (0,0) and the foci are on the x-axis and y-axis, then we can easily derive the ellipse equation. The equation of the ellipse is given by; x2/a2 + y2/b2 = 1.

Is a parabola a conic?

Parabola Equations – MathBitsNotebook(Geo – CCSS Math) A parabola is a conic section. It is a slice of a right cone parallel to one side (a generating line) of the cone. Like the circle, the parabola is a quadratic relation, but unlike the circle, either x will be squared or y will be squared, but not both.