Conic Sections Overview

Conic sections are the curves formed by the intersection of a plane with a double-napped cone. Depending on the angle and position of the plane, we get four main types of conic sections:

1. Circle
2. Ellipse
3. Parabola
4. Hyperbola

Each conic section has unique properties and equations that define its shape.

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1. Circle

A circle is obtained when the plane cuts the cone perpendicular to its axis. It is a special case of an ellipse where both axes are equal.

Standard Equation of a Circle

$(x - h)^2 + (y - k)^2 = r^2$ where:

• $(h, k)$ is the center of the circle,
• $r$ is the radius.

Properties:

• Every point on a circle is equidistant from the center.
• The eccentricity of a circle is 0.

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2. Ellipse

An ellipse is formed when the plane cuts the cone at an angle, but does not go parallel to the base.

Standard Equation of an Ellipse

$\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$ where:

• $(h, k)$ is the center,
• $a$ is the semi-major axis,
• $b$ is the semi-minor axis.

Properties:

• The sum of the distances from any point on the ellipse to two fixed points called foci is constant.
• The eccentricity ($e$) is between 0 and 1:
  $e = \frac{c}{a}, \quad c^2 = a^2 - b^2$
• If $a = b$, the ellipse is a circle.

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3. Parabola

A parabola is obtained when the plane is parallel to the slant height of the cone.

Standard Equation of a Parabola

$y - k = a(x - h)^2$ or
$x - h = a(y - k)^2$ depending on the direction of the parabola.

Properties:

• Every point is equidistant from a fixed focus and a fixed directrix.
• The eccentricity ($e$) of a parabola is always 1.
• The axis of symmetry passes through the vertex and focus.

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4. Hyperbola

A hyperbola is formed when the plane cuts both nappes of the cone.

Standard Equation of a Hyperbola

$\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$ or
$\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1$ depending on whether the hyperbola opens horizontally or vertically.

Properties:

• The difference of the distances from any point on the hyperbola to the two foci is constant.
• The eccentricity ($e$) is greater than 1:
  $e = \frac{c}{a}, \quad c^2 = a^2 + b^2$
• Hyperbolas have asymptotes, which guide their shape but are never touched.

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Summary of Conic Sections

Conic	Equation	Eccentricity ($e$)	Key Property
Circle	$(x - h)^2 + (y - k)^2 = r^2$	$0$	All points equidistant from center.
Ellipse	$\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$	$0 < e < 1$	Sum of distances from foci is constant.
Parabola	$y - k = a(x - h)^2$	$1$	Equidistant from focus and directrix.
Hyperbola	$\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$	$e > 1$	Difference of distances from foci is constant.

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Related Questions:

1. How do you derive the equations of conic sections?
2. What are real-life applications of conic sections?
3. How do eccentricity and directrix relate to conic sections?
4. Can you explain the focus-directrix property in more detail?
5. How do conic sections appear in physics and astronomy?

Tip:

If a quadratic equation in $x$ and $y$ appears, you can identify its conic type by completing the square! 🚀