which degenerate forms of the parabola cannot be obtained from the intersection of a plane and a double napped cone. Discribe how to obtain the forms

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thilina-g's profile pic

thilina-g | College Teacher | (Level 1) Educator

Posted on

The degenerate form that you cannot obtain by the intersection is the two parellal lines (or sometimes the straight line)

You can first insert a horizontal plane across the bottom cone, it will give you the circle, which is a degenrate form of parabola,

Then if you tilt the plane upwards you get an elipse if your intesectin plane doesn't cut the bottom of the cone,

Then when it cuts the bottom of the bottom cone you get a parabola and

finally if you tilt further and when the plane cuts the top cone also you get the hyperbola

cosinusix's profile pic

cosinusix | College Teacher | (Level 3) Assistant Educator

Posted on

Look at the family of parabolae y=ax^2 where a is the parameter.

Their graphs are nested into each other. All of them passes through the point (0,0). The smaller a is, the flatter the curve is.

 

A degenerated form would occur when a goes to 0 or to infinity.

 

If a approaches 0, the curve will become very flat. The degenerated curve will be the x-axis. It may be obtained by the intersection of a plane tangent to a double cone. It is not a solution to our problem!

 

Let's try a increasing to infinity. The branches of the parabolae  will be closer an closer to the y-axis. The degenerated curve will be the positive part of the y-axis. It is a half line. It can't be obtained by the intersection of a plane and a double napped cone.

 

 

 

 

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