123 LearnersLast updated on August 5th, 2025
Conic Sections Formulas
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 .
List of Conic Section Formulas
Conic sections include circles, ellipses, parabolas, and hyperbolas. Let’s explore the formulas used to describe these conic sections.
Formula for Circle
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]
Formula for Ellipse
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.
Formula for Parabola
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)
Formula for Hyperbola
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.
Importance of Conic Section Formulas
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.
Common Mistakes and How to Avoid Them While Using Conic Section Formulas
Students make errors when working with conic sections. Here are some mistakes and ways to avoid them.
Mistake 1
Confusing the terms in formulas
Students often mix up terms and variables in different conic section formulas. To avoid this, practice each formula separately and understand the unique properties of each conic.
Mistake 2
Incorrectly identifying the conic section
Mistaking one conic section for another leads to errors. Understand the distinct characteristics of circles, ellipses, parabolas, and hyperbolas to identify them correctly.
Mistake 3
Misplacing the center or vertices
Students sometimes incorrectly place the center or vertices of conic sections. Always double-check the coordinates and ensure they match the given problem.
Mistake 4
Forgetting to square terms
Forgetting to square terms in the formulas leads to incorrect answers. Carefully follow the formula structure and ensure all squared terms are correctly applied.
Mistake 5
Overlooking the signs in the formula
Students often overlook the signs in the formulas, especially in hyperbolas. Pay attention to the plus and minus signs, which dictate the conic's orientation and properties.
Examples of Problems Using Conic Section Formulas
Problem 1
Find the equation of a circle with center (3, -2) and radius 5.
The equation is \((x - 3)^2 + (y + 2)^2 = 25\)
Explanation
Using the formula: \((x - h)^2 + (y - k)^2 = r^2\) Center (h, k) = (3, -2) and radius r = 5. Thus, \((x - 3)^2 + (y + 2)^2 = 25\).
Problem 2
Write the standard form of an ellipse with center (0, 0), a = 4, b = 2.
The equation is \(\frac{x^2}{16} + \frac{y^2}{4} = 1\)
Explanation
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: \(\frac{x^2}{16} + \frac{y^2}{4} = 1\).
Problem 3
Find the equation of a parabola with vertex (0, 0) and focus at (0, 3).
The equation is \(y = \frac{1}{12}x^2\)
Explanation
Since the parabola opens upwards, use: \[y = \frac{1}{4p}x^2\] Here, p = 3, so: \[y = \frac{1}{12}x^2\].
Problem 4
Determine the equation of a hyperbola with center (0, 0), a = 3, b = 4.
The equation is \(\frac{x^2}{9} - \frac{y^2}{16} = 1\)
Explanation
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: \(\frac{x^2}{9} - \frac{y^2}{16} = 1\).
Problem 5
Find the equation of a circle with radius 7 and center at (5, -4).
The equation is \((x - 5)^2 + (y + 4)^2 = 49\)
Explanation
Using the formula: \((x - h)^2 + (y - k)^2 = r^2\) Center (h, k) = (5, -4) and radius r = 7. Thus, \((x - 5)^2 + (y + 4)^2 = 49\).
FAQs on Conic Sections Formulas
1.What is the formula for a circle?
2.What is the standard form equation of an ellipse?
3.How to find the equation of a parabola?
4.What is the equation of a hyperbola?
5.What are the foci of an ellipse?
Glossary for Conic Sections Formulas
- Circle: A set of points equidistant from a center point in a plane.
- Ellipse: A set of points where the sum of distances from two foci is constant.
- Parabola: A set of points equidistant from a focus and a directrix.
- Hyperbola: A set of points where the difference of distances from two foci is constant.
- Foci: Fixed points used to define and describe conic sections.
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Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
