Conjugate Hyperbola
Category : JEE Main & Advanced
The hyperbola whose transverse and conjugate axis are respectively the conjugate and transverse axis of a given hyperbola is called conjugate hyperbola of the given hyperbola.
Difference between both hyperbolas will be clear from the following table :
| 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 | \[x=\pm a/e\] | \[y=\pm b/e\] |
| Eccentricity | \[e=\sqrt{\left( \frac{{{a}^{2}}+{{b}^{2}}}{{{a}^{2}}} \right)}\] | \[e=\sqrt{\left( \frac{{{a}^{2}}+{{b}^{2}}}{{{b}^{2}}} \right)}\] |
| Length of latus rectum | \[2{{b}^{2}}/a\] | \[2{{a}^{2}}/b\] |
| Parametric co-ordinates | \[(a\sec \phi ,\,b\tan \phi )\] \[0\le \phi <2\pi \] | \[(b\,\sec \phi ,\,a\tan \phi )\] \[0\le \phi <2\pi \] |
| Focal radii | \[SP=e{{x}_{1}}-a\] \[{S}'P=e{{x}_{1}}+a\] | \[SP=e{{y}_{1}}-b\] \[{S}'P=e{{y}_{1}}+b\] |
| Difference of focal radii \[({S}'P-SP)\] | \[2a\] | \[2b\] |
| Tangents at the vertices | \[x=-a,\,x=a\] | \[y=-b,\,y=b\] |
| Equation of the transverse axis | \[y=0\] | \[x=0\] |
| Equation of the conjugate axis | \[x=0\] | \[y=0\] |
You need to login to perform this action.
You will be redirected in
3 sec
