\displaystyle \textbf{1. Definition of a Hyperbola}
\displaystyle \text{A hyperbola is the locus of a point in a plane which moves in such a way that the ratio of its}
\displaystyle \text{distance from a fixed point (called focus) to its distance from a fixed line (called}
\displaystyle \text{directrix) is always constant and greater than unity.}
\displaystyle \text{The fixed point is called the focus, the fixed line is called the directrix and the constant ratio,}
\displaystyle \text{generally denoted by }e\text{, is known as the eccentricity of the hyperbola.}
\displaystyle \text{The general equation of the hyperbola is of the form}
\displaystyle ax^2+2hxy+by^2+2gx+2fy+c=0
\displaystyle \text{where}
\displaystyle abc+2fgh-af^2-bg^2-ch^2\neq0\quad\text{and}\quad h^2>ab.
\\

\displaystyle \textbf{2. Standard Hyperbola } \frac{x^2}{a^2}-\frac{y^2}{b^2}=1
\displaystyle \text{Centre: }(0,0)
\displaystyle \text{Vertices: }(a,0)\text{ and }(-a,0)
\displaystyle \text{Foci: }(\pm ae,0)
\displaystyle \text{Length of transverse axis: }2a
\displaystyle \text{Length of conjugate axis: }2b
\displaystyle \text{Directrices: }x=\pm\frac{a}{e}
\displaystyle \text{Eccentricity: }e=\sqrt{\frac{a^2+b^2}{a^2}}
\displaystyle \text{or }b^2=a^2(e^2-1)
\displaystyle \text{Length of latus-rectum: }\frac{2b^2}{a}
\displaystyle \text{Equation of transverse axis: }y=0
\displaystyle \text{Equation of conjugate axis: }x=0
\\

\displaystyle \textbf{3. Conjugate Hyperbola } -\frac{x^2}{a^2}+\frac{y^2}{b^2}=1
\displaystyle \text{Centre: }(0,0)
\displaystyle \text{Vertices: }(0,b)\text{ and }(0,-b)
\displaystyle \text{Foci: }(0,\pm be)
\displaystyle \text{Length of transverse axis: }2b
\displaystyle \text{Length of conjugate axis: }2a
\displaystyle \text{Directrices: }y=\pm\frac{b}{e}
\displaystyle \text{Eccentricity: }e=\sqrt{\frac{b^2+a^2}{b^2}}
\displaystyle \text{or }a^2=b^2(e^2-1)
\displaystyle \text{Length of latus-rectum: }\frac{2a^2}{b}
\displaystyle \text{Equation of transverse axis: }x=0
\displaystyle \text{Equation of conjugate axis: }y=0
\\

\displaystyle \textbf{4. Conjugate Hyperbola}
\displaystyle \text{A hyperbola whose transverse and conjugate axes are respectively the conjugate and}
\displaystyle \text{transverse axes of a given hyperbola is called the conjugate hyperbola}
\displaystyle \text{of the given hyperbola.}
\\

\displaystyle \textbf{5. Hyperbola with Centre } (h,k)
\displaystyle \text{If the centre of the hyperbola is at the point }(h,k)\text{ and the directions of the axes are parallel}
\displaystyle \text{to the coordinate axes, then its equation is}
\displaystyle \frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1
\\

Discover more from ICSE / ISC / CBSE Mathematics Portal for K12 Students

Subscribe to get the latest posts sent to your email.