HST_1101: PRE-CALCULUS
Conic Sections, Arithmetic and Geometric Sequence and Series, and Trigonometry
CONIC SECTIONS
During the initial quarter of our inaugural semester, our focus was centered on conic sections. A conic section is a curved shape resulting from the intersection of a plane with the surface of a cone. There are three primary types of conic sections: the hyperbola, the parabola, and the ellipse. Additionally, the circle is regarded as a special case of the ellipse, and at times, it was referred to as a distinct fourth type.
CIRCLE
A circle represents a conic section formed by a curved line connecting a set of points that share an equal distance from the central point. The plane intersecting this circle is parallel to the base of the cone.
The collection of all points within the Cartesian plane maintains a consistent distance, referred to as the radius (r), from a fixed point identified as the center (C).
In standard form, the equation takes the structure Ax^2 + Ay^2 + Dx + Ey - F = 0, where both x and y consistently have a coefficient of 1. Conversely, in the general form, expressed as (x - h)^2 + (y - k)^2 = r^2, there are no restrictions on the coefficients of x and y, allowing A to take any value other than 1.
ELLIPSE
An ellipse is defined as the locus of all points within an XY-plane, where the sum of the distances from each point to two fixed points, referred to as foci, remains constant.
As one of the conic sections, an ellipse emerges when a plane intersects a cone at an angle with its base. In the event that the cutting plane runs parallel to the base of the cone, the resulting shape is a circle.
PARABOLA
Parabolas, with their distinctive U shaped curves, are widely encountered in various mathematical and real world applications.
GENERAL EQUATION OF THE PARABOLA
General Equation of a parabola: The standard form of equation for a parabola: The standard form of the equation for a parabola with it's vertex at the origin in Pre calculus is y=ax^2
VERTEX
In the context of a parabola, the vertex refers to a specific point on the curve where the parabola changes direction. It is the highest or lowest point on the graph of the parabolic function, depending on whether the parabola opens upward or downward.
AXIS OF SYMMETRY
The axis of symmetry of a graph or geometric shape is a line that divides the shape into two symmetric halves that are mirror images of each other.In the context of a parabola, the axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two symmetric parts. It is equidistant from both sides of the parabola.
FOCUS
In a parabola, the focus is a fixed point that holds a significant geometric relationship with the parabola. The focus lies on the axis of symmetry and is located inside the curve at an equal distance from the directrix (a line outside the curve) as any point on the parabola. Mathematically, the focus of a parabola is a point that has a specific relationship with respect to the parabola's directrix and is defined in terms of its distance from the directrix. For a parabola defined by the equation
LATUS RECTUM
In a parabola, the latus rectum is a line segment that passes through the focus of the parabola and is perpendicular to the axis of symmetry. It is a specific chord of the parabola that has endpoints on the parabola itself. The length of the latus rectum is determined by the distance from the focus to a point on the parabola that is parallel to the axis of symmetry and passes through the directrix.
DIRECTRIX
In geometry, specifically in the context of a parabola, the directrix is a fixed straight line that is located outside the parabola. It is a line that plays a crucial role in defining the shape and properties of the parabolic curve.
The directrix is always perpendicular to the axis of symmetry of the parabola and is positioned at an equal distance from the vertex as the focus of the parabola but on the opposite side.
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