Formula Sheet: Conic Sections
This chapter explores conic sections including circles, ellipses, parabolas, and hyperbolas, highlighting their definitions and significance in mathematics and real-world applications.
Conic Sections – Formula & Equation Sheet
Essential formulas and equations from Mathematics, tailored for Class 11.
This one-pager compiles key formulas and equations from the Conic Sections chapter of Mathematics. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Formula sheet
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
Circle: (x - h)² + (y - k)² = r²
Where (h, k) is the center and r is the radius. This formula represents all points equidistant from the center, defining a circle in a Cartesian plane.
Ellipse: (x²/a²) + (y²/b²) = 1
Where 2a and 2b are the lengths of the major and minor axes, respectively. This describes an ellipse centered at the origin with foci along the x or y-axis.
Parabola: y² = 4ax
Where a is the distance from the vertex to the focus. This equation describes a parabola that opens rightward with the vertex at the origin.
Hyperbola: (x²/a²) - (y²/b²) = 1
With the center at the origin, this equation defines a hyperbola with foci along the x-axis, where a is the distance from the center to a vertex.
Latus Rectum of a Parabola: Length = 4a
The latus rectum is the width of the parabola at the focus. It measures how wide the parabola opens.
Eccentricity of an Ellipse: e = c/a
Where c is the distance from the center to a focus. This ratio describes the deviation of the ellipse from being circular.
Eccentricity of a Hyperbola: e = c/a
c is the distance from the center to a focus. For hyperbolas, this ratio is always greater than 1.
Length of Latus Rectum of an Ellipse: Length = (2b²/a)
This measures how wide the ellipse opens at the foci.
Length of Latus Rectum of a Hyperbola: Length = (2b²/a)
This describes the distance across the hyperbola at its foci.
Standard Form of Circle: x² + y² = r² (center at origin)
This is a specific case of the circle equation with center at the origin.
Equations
For Circle: x² + y² + 2gx + 2fy + c = 0
General form of the circle's equation in which g, f, and c can be computed from specific points.
For Ellipse: (x²/a²) + (y²/b²) = 1 (major axis along x-axis)
Defines the relationship of distances from the foci to any point on the ellipse.
For Parabola: y² = -4ax (opens left)
This describes a parabola with its vertex at the origin and opens to the left.
Hyperbola in Standard Form: (y²/a²) - (x²/b²) = 1
Defines a hyperbola with a vertical transverse axis, where a determines the distance to the vertices along the y-axis.
Eccentricity of an Ellipse: e = √(1 - b²/a²)
Here, e measures ellipticity, with a being the semi-major axis and b the semi-minor axis.
Eccentricity of a Hyperbola: e = √(1 + b²/a²)
Eccentricity is computed based on the transverse and conjugate axes.
Directrix of Parabola: y = a (for horizontal parabolas)
This line serves as a reference from which points on the parabola are equidistant to the directrix and the focus.
Equation of a Hyperbola: xy/c² = 1 (standard orientation)
Indicates a rectangular hyperbola; c is the distance from the center to the foci.
Foci of an Ellipse: (±c, 0) where c = √(a² - b²)
Locates the foci based on the major and minor axes.
Coordinates of the vertex in a Parabola: (0, 0) (when centered at origin)
Indicates the point that is closest to the focus in standard forms.
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