Let's consider a cone defined by x2 + y? - z² = 0.Then, by intersecting the cone with planes, we can obtain several planecurves defined by quadrics.It is called conic sections (or simply conic).For example, a circle is a conic because it can be obtained by intersectingthe cone x + y- z = 0 with a plane z = c for non-zero constant c.Question: Find equation of a plane which produces a parabola byintersecting with the cone x +y²22 = 0. Note that if a plane is parallel to the generating line of a cone, then we canobtain parabola by intersecting the plane with the cone.Figure:circleellipseparabolahyperbola(Reference: "Conic section" from Wikipedia, the free encyclopedia.)To find a generating line, let's consider intersection of yz-plane x = 0 withthe cone r + y - z2 = 0. Then, we obtain two generating lines of thecone in yz-plane: z = y and z = -y.Now, our goal is to find a plane which is parallel to the generating line (sayz= y) of the cone which does not contain the origin (0, 0, 0). (If the planecontain the origin and parallel to the generating line, then the intersection ofthe plane with the cone will be just the generating line.)

The planar conic sections can be obtained by cutting the cone with a plane. The intersection of the…

.2 Let's consider a cone defined by x + y? – z² = 0. Then, by intersecting the cone with planes, we can obtain several plane curves defined by quadrics. It is called conic sections (or simply conic). For example, a circle is a conic because it can be obtained by intersecting the cone x2 + y? z- = 0 with a plane z = c for non-zero constant C. Question: Find equation of a plane which produces a parabola by intersecting with the cone x+ y²- z = 0.

.2 Let's consider a cone defined by x + y? – z² = 0. Then, by intersecting the cone with planes, we can obtain several plane curves defined by quadrics. It is called conic sections (or simply conic). For example, a circle is a conic because it can be obtained by intersecting the cone x2 + y? z- = 0 with a plane z = c for non-zero constant C. Question: Find equation of a plane which produces a parabola by intersecting with the cone x+ y²- z = 0.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Let's consider a cone defined by x2 + y? - z² = 0.
Then, by intersecting the cone with planes, we can obtain several plane
curves defined by quadrics.
It is called conic sections (or simply conic).
For example, a circle is a conic because it can be obtained by intersecting
the cone x + y- z = 0 with a plane z = c for non-zero constant c.
Question: Find equation of a plane which produces a parabola by
intersecting with the cone x +y²
22 = 0.

Note that if a plane is parallel to the generating line of a cone, then we can
obtain parabola by intersecting the plane with the cone.
Figure:
circle
ellipse
parabola
hyperbola
(Reference: "Conic section" from Wikipedia, the free encyclopedia.)
To find a generating line, let's consider intersection of yz-plane x = 0 with
the cone r + y - z2 = 0. Then, we obtain two generating lines of the
cone in yz-plane: z = y and z = -y.
Now, our goal is to find a plane which is parallel to the generating line (say
z= y) of the cone which does not contain the origin (0, 0, 0). (If the plane
contain the origin and parallel to the generating line, then the intersection of
the plane with the cone will be just the generating line.)

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