What conic section can be formed when a plane cuts the cone parallel to the base of the cone?

A conic section is the intersection of a plane and a double right circular cone . By changing the angle and location of the intersection, we can produce different types of conics. There are four basic types: circles , ellipses , hyperbolas and parabolas . None of the intersections will pass through the vertices of the cone.

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STANDARD FORMS OF EQUATIONS OF CONIC SECTIONS:
Solving Systems of Equations
Which conic section do you get when you cut the cone parallel to its axis?
When a right circular cone is cut by a plane parallel to the axis the section view will be obtained?
When a cone is cut by planes at different angles the curves of intersection are called?
What happens when a plane intersects a cone?
When the plane intersects the cone exactly at its vertex?
What is the curve formed when a plane cuts a cone?
What happens when you cut a cone at different angles?

If the right circular cone is cut by a plane perpendicular to the axis of the cone, the intersection is a circle. If the plane intersects one of the pieces of the cone and its axis but is not perpendicular to the axis, the intersection will be an ellipse. To generate a parabola, the intersecting plane must be parallel to one side of the cone and it should intersect one piece of the double cone. And finally, to generate a hyperbola the plane intersects both pieces of the cone. For this, the slope of the intersecting plane should be greater than that of the cone.

The general equation for any conic section is

A x 2 + B x y + C y 2 + D x + E y + F = 0 where A , B , C , D , E and F are constants.

As we change the values of some of the constants, the shape of the corresponding conic will also change. It is important to know the differences in the equations to help quickly identify the type of conic that is represented by a given equation.
      If B 2 − 4 A C is less than zero, if a conic exists, it will be either a circle or an ellipse.
      If B 2 − 4 A C equals zero, if a conic exists, it will be a parabola.
      If B 2 − 4 A C is greater than zero, if a conic exists, it will be a hyperbola.

STANDARD FORMS OF EQUATIONS OF CONIC SECTIONS:

Circle	( x − h ) 2 + ( y − k ) 2 = r 2	Center is ( h , k ) . Radius is r .
Ellipse with horizontal major axis	( x − h ) 2 a 2 + ( y − k ) 2 b 2 = 1	Center is ( h , k ) . Length of major axis is 2 a . Length of minor axis is 2 b . Distance between center and either focus is c with c 2 = a 2 − b 2 , a > b > 0 .
Ellipse with vertical major axis	( x − h ) 2 b 2 + ( y − k ) 2 a 2 = 1	Center is ( h , k ) . Length of major axis is 2 a . Length of minor axis is 2 b . Distance between center and either focus is c with c 2 = a 2 − b 2 , a > b > 0 .
Hyperbola with horizontal transverse axis	( x − h ) 2 a 2 − ( y − k ) 2 b 2 = 1	Center is ( h , k ) . Distance between the vertices is 2 a . Distance between the foci is 2 c . c 2 = a 2 + b 2
Hyperbola with vertical transverse axis	( y − k ) 2 a 2 − ( x − h ) 2 b 2 = 1	Center is ( h , k ) . Distance between the vertices is 2 a . Distance between the foci is 2 c . c 2 = a 2 + b 2
Parabola with horizontal axis	( y − k ) 2 = 4 p ( x − h ) , p ≠ 0	Vertex is ( h , k ) . Focus is ( h + p , k ) . Directrix is the line x = h − p Axis is the line y = k
Parabola with vertical axis	( x − h ) 2 = 4 p ( y − k ) , p ≠ 0	Vertex is ( h , k ) . Focus is ( h , k + p ) . Directrix is the line y = k − p . Axis is the line x = h

Solving Systems of Equations

You must be familiar with solving system of linear equation . Geometrically it gives the point(s) of intersection of two or more straight lines. In a similar way, the solutions of system of quadratic equations would give the points of intersection of two or more conics.

Algebraically a system of quadratic equations can be solved by elimination or substitution just as in the case of linear systems.

Example:

Solve the system of equations.

x 2 + 4 y 2 = 16 x 2 + y 2 = 9

The coefficient of x 2 is the same for both the equations. So, subtract the second equation from the first to eliminate the variable x . You get:

3 y 2 = 7

Solving for y :

3 y 2 3 = 7 3 y 2 = 7 3 y = ± 7 3

Use the value of y to evaluate x .

x 2 + 7 3 = 9 x 2 = 9 − 7 3 = 20 3 x = ± 20 3

Therefore, the solutions are ( + 20 3 , + 7 3 ) , ( + 20 3 , − 7 3 ) , ( − 20 3 , + 7 3 ) and ( − 20 3 , − 7 3 ) .

Now, let us look at it from a geometric point of view.

If you divide both sides of the first equation x 2 + 4 y 2 = 16 by 16 you get x 2 16 + y 2 4 = 1 . That is, it is an ellipse centered at origin with major axis 4 and minor axis 2 . The second equation is a circle centered at origin and has a radius 3 . The circle and the ellipse meet at four different points as shown.

Explanation: When the plane cuts the cone at an angle parallel to the axis of the cone, then the resulting conic section is called as a hyperbola. If the plane cuts the cone at an angle with respect to the axis of the cone then the resulting conic sections are called as ellipse and parabola. 7.

Which conic section do you get when you cut the cone parallel to its axis?

Cutting parallel to a side of the cone produces a parabola. Cutting more nearly parallel to the axis than to the side produces a hyperbola (the hyperbola in the diagram represents a cut parallel to the axis of the cone). View from above of, from left to right, a circle, an ellipse, a parabola and a hyperbola.

When a plane intersects a cone at an angle that is parallel to the edge of the cone?

A parabola is formed when a plane intersects a cone and that plane lies parallel to the edge of that cone.

What conic section is formed when a cone is cut by a plane parallel to the vertical axis?

hyperbola
Conic sections are generated by the intersection of a plane with a cone. If the plane is parallel to the axis of revolution (the y -axis), then the conic section is a hyperbola.

When a right circular cone is cut by a plane parallel to the axis the section view will be obtained?

Explanation: If a cone made to cut by a plane parallel to its axis and some distance away from it the section formed is hyperbola. If the section plane is perpendicular to axis the section is circle. If section plane passes through apex the section formed is a triangle.

When a cone is cut by planes at different angles the curves of intersection are called?

Answer: In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane.

When a cone is cut by a plane perpendicular?

When we are cutting the cone parallel to its and generator then which of the following curve will be generated at the cutting surface?

Explanation: When the plane cuts the cone parallel to the generator the curve traced out is Parabola.

What happens when a plane intersects a cone?

If you intersect a cone with a plane so that the plane is parallel to the base, you get a circle. A circle is defined as the set of all points whose distance from a fixed point (the center) is always the same.

When the plane intersects the cone exactly at its vertex?

Point: If the plane intersects the two cones at the vertex and at an angle greater than the vertex angle, we get a point. This is a degenerate ellipse. Line: If the plane intersects the two cones at the vertex and at an angle equal to the vertex angle, we get a line. This is a degenerate parabola.

What can be formed when the plane is cutting parallel to the base?

A circle is the conic section formed when the cutting plane is parallel to the base of the cone or equivalently perpendicular to the axis.

When a right circular cone is cut by a plane?

A right circular cone is cut by a plane parallel to its base in such a way that the slant heights of the original and the smaller cone thus obtained are in the ratio 2:1.

What is the curve formed when a plane cuts a cone?

When the plane cuts the cone at an angle between a perpendicular to the axis (which would produce a circle) and an angle parallel to the side of the cone (which would produce a parabola), the curve formed is an ellipse.

What happens when you cut a cone at different angles?

Angled view of a cone, with conic sections produced by cutting the cone at different angles. Cutting at right angles to the axis produces a circle. Cutting at less than a right angle to the axis but more than the angle made by the side of the cone produces an ellipse.

What is the name of the conics formed if the plane?

If the plane cuts at an angle to the axis but does not cut all the generators then what is the name of the conics formed? Explanation: If the plane cuts at an angle with respect to the axis and does not cut all the generators then the conics formed is a parabola.

What is the conic section of a cone called?