Last updated on August 2nd, 2025
In mathematics, conic sections are the curves obtained by intersecting a right circular cone with a plane. These include the circle, ellipse, parabola, and hyperbola. In this topic, we will learn the formulas related to conic sections as covered in .
Conic sections include circles, ellipses, parabolas, and hyperbolas. Let’s explore the formulas used to describe these conic sections.
A circle is a set of all points in a plane that are a fixed distance from a given point, the center. The standard equation of a circle with center (h, k) and radius r is:
[(x - h)^2 + (y - k)^2 = r^2]
An ellipse is the set of all points for which the sum of the distances from two fixed points (foci) is constant. The standard equation of an ellipse centered at (h, k) is:
[\frac{(x - h)^2}{a^2} + frac{(y - k)^2}{b^2} = 1] where a > b for a horizontal ellipse and b > a for a vertical ellipse.
A parabola is the set of all points in the plane that are equidistant from a fixed point (focus) and a given line (directrix). The standard equation of a parabola with vertex at (h, k) is:
[y - k = a(x - h)^2] (for a vertical parabola) [x - h = a(y - k)^2] (for a horizontal parabola)
A hyperbola is the set of all points where the difference of the distances from two fixed points (foci) is constant. The standard equation of a hyperbola centered at (h, k) is:
[frac{(x - h)^2}{a^2} - frac{(y - k)^2}{b^2} = 1] for a horizontal hyperbola, and [frac{(y - k)^2}{a^2} - frac{(x - h)^2}{b^2} = 1] for a vertical hyperbola.
Conic sections are fundamental in mathematics and have numerous real-life applications.
These formulas help describe planetary orbits, design optical lenses, and analyze satellite paths.
By learning these formulas, students can understand key concepts in geometry, physics, and engineering.
Students make errors when working with conic sections. Here are some mistakes and ways to avoid them.
Find the equation of a circle with center (3, -2) and radius 5.
The equation is \((x - 3)^2 + (y + 2)^2 = 25\)
Using the formula: \((x - h)^2 + (y - k)^2 = r^2\) Center (h, k) = (3, -2) and radius r = 5. Thus, \((x - 3)^2 + (y + 2)^2 = 25\).
Write the standard form of an ellipse with center (0, 0), a = 4, b = 2.
The equation is \(\frac{x^2}{16} + \frac{y^2}{4} = 1\)
For an ellipse centered at the origin with a > b: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) Here, a = 4, b = 2, so: \(\frac{x^2}{16} + \frac{y^2}{4} = 1\).
Find the equation of a parabola with vertex (0, 0) and focus at (0, 3).
The equation is \(y = \frac{1}{12}x^2\)
Since the parabola opens upwards, use: \[y = \frac{1}{4p}x^2\] Here, p = 3, so: \[y = \frac{1}{12}x^2\].
Determine the equation of a hyperbola with center (0, 0), a = 3, b = 4.
The equation is \(\frac{x^2}{9} - \frac{y^2}{16} = 1\)
For a hyperbola centered at the origin with a horizontal transverse axis: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) Here, a = 3, b = 4, so: \(\frac{x^2}{9} - \frac{y^2}{16} = 1\).
Find the equation of a circle with radius 7 and center at (5, -4).
The equation is \((x - 5)^2 + (y + 4)^2 = 49\)
Using the formula: \((x - h)^2 + (y - k)^2 = r^2\) Center (h, k) = (5, -4) and radius r = 7. Thus, \((x - 5)^2 + (y + 4)^2 = 49\).
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