Examples of half-plane in the following topics:
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- A straight line drawn through the plane divides the plane into two half-planes, as shown in the diagram below.
- Next, choose a test point to figure out which half-plane we need to shade in.
- This a true statement, so shade in the half-plane containing $(0, 0). $
- All points in the shaded half-plane above the line are solutions to this inequality.
- The boundary line shown above divides the coordinate plane into two half-planes.
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- Named for "the father of analytic geometry," 17th-century French mathematician René Descartes, the Cartesian coordinate system is a 2-dimensional plane with a horizontal axis and a vertical axis.
- Points are specified uniquely in the Cartesian plane by a pair of numerical coordinates, which are the signed distances from the point to the two axes.
- Therefore, you move one and a half units left and two and a half units down.
- A Cartesian plane is particularly useful for plotting a series of points that show a relationship between two variables.
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- In other words, the plane is at the same angle as the outside surface of the cone.
- The axis of symmetry is a line that is at the same angle as the cone and divides the parabola in half.
- The vertex is the point where the plane intersects the exterior surface of the cone.
- On a coordinate plane, parabolas are frequently encountered as graphs of quadratic functions, such as:
- This diagram shows how a parabola is generated by the intersection of a plane with a right circular cone.
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- In mathematics, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve .
- Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis.
- The semi-major axis (denoted by a in the figure) and the semi-minor axis (denoted by b in the figure) are one half of the major and minor axes, respectively.
- The area enclosed by an ellipse is $\pi ab$, where a and b are one-half of the ellipse's major and minor axes respectively.
- One definition of an ellipse is the intersection of a cone with an inclined plane.
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- The set of all points in a plane that are the same distance from a given point forms a circle.
- The set of all points in a plane that are the same distance from a given point forms a circle.
- Radius: a line segment joining the center of the circle to any point on the circle itself; or the length of such a segment, which is half a diameter.
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- Graphically, the solutions fall on a line or plane
that is the intersection of three planes in space.
- Notice that
two of the planes are the same, and they intersect the third plane on a
line.
- The equations could
represent three parallel planes, two parallel planes and one
intersecting plane, or three planes that intersect the other two but not
at the same location.
- (b) Two of the planes are parallel and intersect with the third plane,
but not with each other.
- Two equations represent the same plane, and these intersect the third plane on a line.
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- Conic sections can be generated by
intersecting a plane with a cone.
- Conic sections are generated by the intersection of a plane with a cone.
- If the plane is parallel to the generating line, the
conic section is a parabola.
- If the plane is perpendicular to the axis
of revolution, the conic section is a circle.
- Each conic is determined by the angle the plane makes with the axis of the cone.
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- Depending on the angle between the plane and the cone, four different intersection shapes can be formed.
- A parabola is formed when the plane is parallel to the surface of the cone, resulting in a U-shaped curve that lies on the plane.
- A circle is formed when the plane is parallel to the base of the cone.
- On a coordinate plane, the general form of the equation of the circle is
- This happens when the plane intersects the apex of the double cone.
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- The time it takes for a substance (drug, radioactive nuclide, or other)
to lose half of its pharmacological, physiological, biological, or
radiological activity is called its half-life.
- Then in $5$ years half the amount ($50$ pounds) remains.
- C-13 has a half-life of 5700 years—that is, in 5700 years, half of a sample of C-13 will have converted to C-12, which represents approximately all the remaining carbon.
- Using the graph, find that half-life.
- Since there is 50% of the substance left after 1 year, the half-life is 1 year.
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- Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part.
- Thus, for example, complex number $-2+3i$ would be associated with the point $(-2,3)$ and would be plotted in the complex plane as shown below.
- The complex number
$-2+3i$ is plotted in the complex plane, $2$ to the left on the real axis, and $3$ up on the imaginary axis.