Hyperbola | \[\frac{{{x}^{\mathbf{2}}}}{{{a}^{\mathbf{2}}}}-\frac{{{y}^{\mathbf{2}}}}{{{b}^{\mathbf{2}}}}=\mathbf{1}\] | \[-\frac{{{x}^{\mathbf{2}}}}{{{a}^{\mathbf{2}}}}+\frac{{{y}^{\mathbf{2}}}}{{{b}^{\mathbf{2}}}}=\mathbf{1}\] or \[\frac{{{x}^{\mathbf{2}}}}{{{a}^{\mathbf{2}}}}-\frac{{{y}^{\mathbf{2}}}}{{{b}^{\mathbf{2}}}}=-\mathbf{1}\] |
Imp. terms | ||
Centre | \[(0,\,\,0)\] | \[(0,\,\,0)\] |
Length of transverse axis | \[2a\] | \[2b\] |
Length of conjugate axis | \[2b\] | \[2a\] |
Foci | \[(\pm \,ae,\,0)\] | \[(0,\,\pm be)\] |
Equation of directrices | \[x=\pm a/e\] | \[y=\pm b/e\] |
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If the centre of hyperbola is \[(h,\,\,k)\] and axes are parallel to the co-ordinate axes, then its equation is \[\frac{{{(x-h)}^{2}}}{{{a}^{2}}}-\frac{{{(y-k)}^{2}}}{{{b}^{2}}}=1\].
Let \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] be the hyperbola, then equation of the auxiliary circle is \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\].
Let \[\angle QCN=\varphi \]. Here P and Q are the corresponding points on the hyperbola and the auxiliary circle \[(0\le \varphi <2\pi )\].
The equations \[x=a\sec \varphi \] and \[y=b\tan \varphi \] are known as the parametric equations of the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\]. This \[(a\sec \varphi ,\,b\tan \varphi )\] lies on the hyperbola for all values of \[\varphi \].
Let the hyperbola be \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\].
Then \[P({{x}_{1}},\,{{y}_{1}})\] will lie inside, on or outside the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] according as \[\frac{x_{1}^{2}}{{{a}^{2}}}-\frac{y_{1}^{2}}{{{b}^{2}}}-1\] is positive, zero or negative.
The straight line \[y=mx+c\] will cut the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] in two points may be real, coincident or imaginary according as \[{{c}^{2}}>,\,=,\,<{{a}^{2}}{{m}^{2}}-{{b}^{2}}\].
Condition of tangency : If straight line \[y=mx+c\] touches the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\], then \[{{c}^{2}}={{a}^{2}}{{m}^{2}}-{{b}^{2}}\].
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