Conic Sections

CONIC SECTIONS EXPLAINED: When a plane slices through a cone at varying angles, we derive curves known as conic sections. There exist four distinct types: the circle, ellipse, parabola, and hyperbola. A deep exploration of their properties and equations takes place in the realms of algebra and geometry. Beyond the classroom, these conic sections have a multitude of practical applications spanning fields such as physics, engineering, and astronomy, to name just a few.

Identifying Conic Sections

Identify the type of the conic section represented by the equation

x^{2} - 4 y^{2} - 2 x + 8 y - 1 = 0

.

Identifying Circles

Identify the center and radius of the circle given by the equation

(x - 3)^{2} + (y + 2)^{2} = 25

.

Finding a Circle Using the Center and Another Point

Given a circle with a center at point A(-1, 3) and another point on the circle B(1, 1), find the equation of the circle.

Finding a Circle by the Diameter End Points

Find the standard form of the equation of the circle with endpoints of a diameter at the points

(5, 2)

and

(- 1, 6)

.

Finding the Parabola Equation Using the Vertex and Another Point

Given the vertex of a parabola is at the point (2,3) and the parabola passes through the point (1,-1), find the equation of the parabola.

Finding the Properties of the Parabola

Select the equations for the directrix lines for the hyperbola defined by the equation

\frac{x^{2}}{20} - \frac{y^{2}}{10} = 1

Finding the Vertex

Find the vertex of the parabola given by the equation

y = 2 x^{2} + 8 x + 7

Finding the Vertex Form of the Parabola

Given the equation of the parabola

y = 2 x^{2} - 4 x + 3

, find the vertex form of the parabola.

Finding the Vertex Form of an Ellipse

Given the equation of an ellipse,

\frac{(x - 5)^{2}}{16} + \frac{(y + 3)^{2}}{9} = 1

, find the vertex form of the ellipse.

Finding the Vertex Form of a Circle

Find the vertex form of the circle with the equation

x^{2} + y^{2} - 6 x + 8 y + 9 = 0

.

Finding the Vertex Form of a Hyperbola

Find the vertex form of the hyperbola given by the equation

16 x^{2} - 9 y^{2} = 144

.

Finding the Standard Form of a Parabola

Find the standard form of the equation of the parabola satisfying the given conditions. Focus:

(- 1, 6)

; Directrix:

y = 2

Finding the Expanded Form of an Ellipse

Find the expanded form of an ellipse with a center at (3, -2), a horizontal semi-axis of 5 units, and a vertical semi-axis of 3 units.

Finding the Expanded Form of a Circle

Find the expanded form of the circle equation with center at point

(h, k) = (- 3, 4)

and radius

r = 5

Finding the Expanded Form of a Hyperbola

Find the expanded form of the hyperbola with the following properties: center at (4, -3), vertices (4, 1) and (4, -7), and foci at (4, 2) and (4, -8).

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