Recognisation of Conics
Category : JEE Main & Advanced
The equation of conics is represented by the general equation of second degree \[a{{x}^{2}}+2hxy+b{{y}^{2}}+2gx+2fy+c=0\] ......(i) and discriminant of above equation is represented by \[\Delta \], where \[\Delta =abc+2fgh-a{{f}^{2}}-b{{g}^{2}}-c{{h}^{2}}\]
Case I : When \[\Delta =0\].
In this case equation (i) represents the degenerate conic whose nature is given in the following table.
| S. No. | Condition | Nature of conic |
| 1. | \[\Delta =0\] and \[ab-{{h}^{2}}=0\grave{\ }\] | A pair of coincident straight lines |
| 2. | \[\Delta =0\] and \[ab-{{h}^{2}}<0\] | A pair of intersecting straight lines |
| 3. | \[\Delta =0\] and \[ab-{{h}^{2}}>0\] | A point |
Case II : When \[\Delta \ne 0\].
In this case equation (i) represents the non-degenerate conic whose nature is given in the following table.
| S. No. | Condition | Nature of conic |
| 1. | \[\Delta \ne 0,\,\,h=0,\,\,a=b,\,\,e=0\] | A circle |
| 2. | \[\Delta \ne 0,\,\,ab-{{h}^{2}}=0,\,\,e=1\] | A parabola |
| 3. | \[\Delta \ne 0,\,\,ab-{{h}^{2}}>0,\,\,e<1\] | An ellipse |
| 4. | \[\Delta \ne 0,\,\,ab-{{h}^{2}}<0,\,\,e>1\] | A hyperbola |
| 5. | \[\Delta \ne 0,\,\,ab-{{h}^{2}}<0,\]\[a+b=0,\,\,e=\sqrt{2}\] | A rectangular hyperbola |
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