If the focus is `(alpha, beta )`, directrix is ax+ by+ c = 0 and P is any point. Then, ratio of distance of point P from the fixed M point (focus) to fixed line (directrix) is
`(PS)/(PM) = e`
`=> sqrt ((x -alpha)^2 + ( y - beta )^2 ) = e ( ax +by +c)/sqrt(a^2 +b^2 )`
`=> (x -alpha)^2 + ( y -beta)^2 = (e^2(ax +by +c ) ^2)/(a^2 +b^2)`
On simplifying above equation, we get second degree equation
`ax^2 + by^2 + 2gx + 2fy + 2hxy + c = 0`
Here discriminant,
`Delta = abc + 2 fgh -af^2 -bg^2 -ch^2`
Equations will represent different conics, they are given
Case I When `Delta = 0`
Condition | Nature of conic |
`ab- h^2 =0` | A pair of coincident lines |
`ab- h^2 < 0` | Real and distinct pair of straight lines |
`ab- h^2 > 0` | Lines are imaginary |
Case II When `Delta != 0`
Condition | Nature of conic |
h =0, a= b, e=0 | A circle |
`ab -h^2 =0 , e =1` | A parabola |
`ab - h^2 > 0 , e < 1` | An ellipse |
`ab - h^2 < 0 , e > 1` | A hyperbola |
`ab -h^2 < 0 , a +b = 0 , e = sqrt2` | A rectangular hyperbola |
If the focus is `(alpha, beta )`, directrix is ax+ by+ c = 0 and P is any point. Then, ratio of distance of point P from the fixed M point (focus) to fixed line (directrix) is
`(PS)/(PM) = e`
`=> sqrt ((x -alpha)^2 + ( y - beta )^2 ) = e ( ax +by +c)/sqrt(a^2 +b^2 )`
`=> (x -alpha)^2 + ( y -beta)^2 = (e^2(ax +by +c ) ^2)/(a^2 +b^2)`
On simplifying above equation, we get second degree equation
`ax^2 + by^2 + 2gx + 2fy + 2hxy + c = 0`
Here discriminant,
`Delta = abc + 2 fgh -af^2 -bg^2 -ch^2`
Equations will represent different conics, they are given
Case I When `Delta = 0`
Condition | Nature of conic |
`ab- h^2 =0` | A pair of coincident lines |
`ab- h^2 < 0` | Real and distinct pair of straight lines |
`ab- h^2 > 0` | Lines are imaginary |
Case II When `Delta != 0`
Condition | Nature of conic |
h =0, a= b, e=0 | A circle |
`ab -h^2 =0 , e =1` | A parabola |
`ab - h^2 > 0 , e < 1` | An ellipse |
`ab - h^2 < 0 , e > 1` | A hyperbola |
`ab -h^2 < 0 , a +b = 0 , e = sqrt2` | A rectangular hyperbola |