Important Conic Sections Formulas for JEE and NEET

 Page 1


Page # 4
CIRCLE
1. Intercepts made by Circle x
2 
+ y
2 
+ 2gx + 2fy + c = 0 on the Axes:
(a) 2 c g
2
? on x -axis (b) 2 c f
2
?  on y - aixs
2. Parametric Equations of a Circle:
x = h + r cos ? ;  y = k + r sin ?
3. Tangent :
(a) Slope form :  y = mx ± 
2
m 1 a ?
(b) Point form :  xx
1
 + yy
1
 = a
2
 or T = o
(c) Parametric form : x cos
 
? + y sin
 
? = a.
4. Pair of Tangents from a Point: SS
1
 = T².
5. Length of a Tangent : Length of tangent is
1
S
6. Director Circle: x
2
 + y
2
 = 2a
2
 for x
2
 + y
2
 = a
2
7. Chord of Contact: T = 0
1. Length of chord of contact =
2 2
L R
R L 2
?
2. Area of the triangle formed by the pair of the tangents & its chord of
contact = 
2 2
3
L R
L R
?
3. Tangent of the angle between the pair of tangents from (x
1
, y
1
)
= 
?
?
?
?
?
?
?
?
?
2 2
R L
L R 2
4. Equation of the circle circumscribing the triangle PT
1 
T
2
 is :
(x ? x
1
) (x + g) + (y ? y
1
) (y + f) = 0.
8. Condition of orthogonality of Two Circles:  2 g
1 
g
2 
+ 2 f
1 
f
2
 = c
1 
+ c
2
.
9. Radical Axis : S
1
 ? S
2
 = 0  i.e. 2
 
(g
1 
? g
2
) x + 2
 
(f
1 
? f
2
) y + (c
1 
? c
2
) = 0.
10. Family of Circles:  S
1
 + K S
2
 = 0, S + KL = 0.
Page 2


Page # 4
CIRCLE
1. Intercepts made by Circle x
2 
+ y
2 
+ 2gx + 2fy + c = 0 on the Axes:
(a) 2 c g
2
? on x -axis (b) 2 c f
2
?  on y - aixs
2. Parametric Equations of a Circle:
x = h + r cos ? ;  y = k + r sin ?
3. Tangent :
(a) Slope form :  y = mx ± 
2
m 1 a ?
(b) Point form :  xx
1
 + yy
1
 = a
2
 or T = o
(c) Parametric form : x cos
 
? + y sin
 
? = a.
4. Pair of Tangents from a Point: SS
1
 = T².
5. Length of a Tangent : Length of tangent is
1
S
6. Director Circle: x
2
 + y
2
 = 2a
2
 for x
2
 + y
2
 = a
2
7. Chord of Contact: T = 0
1. Length of chord of contact =
2 2
L R
R L 2
?
2. Area of the triangle formed by the pair of the tangents & its chord of
contact = 
2 2
3
L R
L R
?
3. Tangent of the angle between the pair of tangents from (x
1
, y
1
)
= 
?
?
?
?
?
?
?
?
?
2 2
R L
L R 2
4. Equation of the circle circumscribing the triangle PT
1 
T
2
 is :
(x ? x
1
) (x + g) + (y ? y
1
) (y + f) = 0.
8. Condition of orthogonality of Two Circles:  2 g
1 
g
2 
+ 2 f
1 
f
2
 = c
1 
+ c
2
.
9. Radical Axis : S
1
 ? S
2
 = 0  i.e. 2
 
(g
1 
? g
2
) x + 2
 
(f
1 
? f
2
) y + (c
1 
? c
2
) = 0.
10. Family of Circles:  S
1
 + K S
2
 = 0, S + KL = 0.
Page # 5
PARABOLA
1. Equation of standard parabola :
y
2
 = 4ax, Vertex is (0, 0),  focus is (a, 0), Directrix is x + a = 0 and Axis is
y = 0. Length of the latus rectum = 4a, ends of the latus rectum are L(a, 2a)
& L’ (a, ? 2a).
2. Parametric Representation: x = at² & y = 2at
3. Tangents to the Parabola y² = 4ax:
1. Slope form y = mx +
m
a
 (m ? 0)2. Parametric form ty = x + at
2
3. Point form T = 0
4. Normals to the parabola y² = 4ax :
y ? y
1
 =
a 2
y
1
?
 (x ? x
1
) at (x
1,
 y
1
)  ; y = mx ? 2am ? am
3
 at (am
2
? ? ? 2am) ;
y + tx = 2at + at
3
 at (at
2
, 2at).
Page 3


Page # 4
CIRCLE
1. Intercepts made by Circle x
2 
+ y
2 
+ 2gx + 2fy + c = 0 on the Axes:
(a) 2 c g
2
? on x -axis (b) 2 c f
2
?  on y - aixs
2. Parametric Equations of a Circle:
x = h + r cos ? ;  y = k + r sin ?
3. Tangent :
(a) Slope form :  y = mx ± 
2
m 1 a ?
(b) Point form :  xx
1
 + yy
1
 = a
2
 or T = o
(c) Parametric form : x cos
 
? + y sin
 
? = a.
4. Pair of Tangents from a Point: SS
1
 = T².
5. Length of a Tangent : Length of tangent is
1
S
6. Director Circle: x
2
 + y
2
 = 2a
2
 for x
2
 + y
2
 = a
2
7. Chord of Contact: T = 0
1. Length of chord of contact =
2 2
L R
R L 2
?
2. Area of the triangle formed by the pair of the tangents & its chord of
contact = 
2 2
3
L R
L R
?
3. Tangent of the angle between the pair of tangents from (x
1
, y
1
)
= 
?
?
?
?
?
?
?
?
?
2 2
R L
L R 2
4. Equation of the circle circumscribing the triangle PT
1 
T
2
 is :
(x ? x
1
) (x + g) + (y ? y
1
) (y + f) = 0.
8. Condition of orthogonality of Two Circles:  2 g
1 
g
2 
+ 2 f
1 
f
2
 = c
1 
+ c
2
.
9. Radical Axis : S
1
 ? S
2
 = 0  i.e. 2
 
(g
1 
? g
2
) x + 2
 
(f
1 
? f
2
) y + (c
1 
? c
2
) = 0.
10. Family of Circles:  S
1
 + K S
2
 = 0, S + KL = 0.
Page # 5
PARABOLA
1. Equation of standard parabola :
y
2
 = 4ax, Vertex is (0, 0),  focus is (a, 0), Directrix is x + a = 0 and Axis is
y = 0. Length of the latus rectum = 4a, ends of the latus rectum are L(a, 2a)
& L’ (a, ? 2a).
2. Parametric Representation: x = at² & y = 2at
3. Tangents to the Parabola y² = 4ax:
1. Slope form y = mx +
m
a
 (m ? 0)2. Parametric form ty = x + at
2
3. Point form T = 0
4. Normals to the parabola y² = 4ax :
y ? y
1
 =
a 2
y
1
?
 (x ? x
1
) at (x
1,
 y
1
)  ; y = mx ? 2am ? am
3
 at (am
2
? ? ? 2am) ;
y + tx = 2at + at
3
 at (at
2
, 2at).
Page # 5
ELLIPSE
1. Standard Equation : 
2
2
2
2
b
y
a
x
? = 1, where a > b & b² = a² (1 ? e²).
Eccentricity:  e =
2
2
a
b
1 ? , (0 < e < 1),  Directrices : x  = ± 
e
a
.
Focii : S ? (± a
 
e, 0). Length of, major axes = 2a and minor axes = 2b
Vertices :   A ? ? ? ( ? a, 0) & A ? (a, 0) .
Latus Rectum : = ? ?
2
2
e 1 a 2
a
b 2
? ?
2. Auxiliary Circle : x² + y² = a²
3. Parametric Representation : x = a cos ? & y = b sin ?
4. Position of a Point w.r.t. an Ellipse:
The point P(x
1,
 y
1
) lies outside, inside or on the ellipse according as;
 1
b
y
a
x
2
2
1
2
2
1
? ? > < or = 0.
5.
The line y = mx + c meets the ellipse 
2
2
2
2
b
y
a
x
? = 1 in two points real,
coincident or imaginary according as c² is < = or > a²m² + b².
6. Tangents:
Slope form: y = mx ± 
2 2 2
a m ? b
, Point form : 1
b
yy
a
xx
2
1
2
1
? ? ,
Parametric form:  1
b
ysin
a
xcos
?
?
?
?
7. Normals:
1
2
1
2
y
b y
x
a x
?
 = a² ? b²,  ax.
 
sec
 
? ? by
. 
cosec
 
? = (a² ? b²), y = mx ?
? ?
2 2 2
2 2
a b m
a b m
?
?
.
8. Director Circle: x² + y² = a² + b²
Page 4


Page # 4
CIRCLE
1. Intercepts made by Circle x
2 
+ y
2 
+ 2gx + 2fy + c = 0 on the Axes:
(a) 2 c g
2
? on x -axis (b) 2 c f
2
?  on y - aixs
2. Parametric Equations of a Circle:
x = h + r cos ? ;  y = k + r sin ?
3. Tangent :
(a) Slope form :  y = mx ± 
2
m 1 a ?
(b) Point form :  xx
1
 + yy
1
 = a
2
 or T = o
(c) Parametric form : x cos
 
? + y sin
 
? = a.
4. Pair of Tangents from a Point: SS
1
 = T².
5. Length of a Tangent : Length of tangent is
1
S
6. Director Circle: x
2
 + y
2
 = 2a
2
 for x
2
 + y
2
 = a
2
7. Chord of Contact: T = 0
1. Length of chord of contact =
2 2
L R
R L 2
?
2. Area of the triangle formed by the pair of the tangents & its chord of
contact = 
2 2
3
L R
L R
?
3. Tangent of the angle between the pair of tangents from (x
1
, y
1
)
= 
?
?
?
?
?
?
?
?
?
2 2
R L
L R 2
4. Equation of the circle circumscribing the triangle PT
1 
T
2
 is :
(x ? x
1
) (x + g) + (y ? y
1
) (y + f) = 0.
8. Condition of orthogonality of Two Circles:  2 g
1 
g
2 
+ 2 f
1 
f
2
 = c
1 
+ c
2
.
9. Radical Axis : S
1
 ? S
2
 = 0  i.e. 2
 
(g
1 
? g
2
) x + 2
 
(f
1 
? f
2
) y + (c
1 
? c
2
) = 0.
10. Family of Circles:  S
1
 + K S
2
 = 0, S + KL = 0.
Page # 5
PARABOLA
1. Equation of standard parabola :
y
2
 = 4ax, Vertex is (0, 0),  focus is (a, 0), Directrix is x + a = 0 and Axis is
y = 0. Length of the latus rectum = 4a, ends of the latus rectum are L(a, 2a)
& L’ (a, ? 2a).
2. Parametric Representation: x = at² & y = 2at
3. Tangents to the Parabola y² = 4ax:
1. Slope form y = mx +
m
a
 (m ? 0)2. Parametric form ty = x + at
2
3. Point form T = 0
4. Normals to the parabola y² = 4ax :
y ? y
1
 =
a 2
y
1
?
 (x ? x
1
) at (x
1,
 y
1
)  ; y = mx ? 2am ? am
3
 at (am
2
? ? ? 2am) ;
y + tx = 2at + at
3
 at (at
2
, 2at).
Page # 5
ELLIPSE
1. Standard Equation : 
2
2
2
2
b
y
a
x
? = 1, where a > b & b² = a² (1 ? e²).
Eccentricity:  e =
2
2
a
b
1 ? , (0 < e < 1),  Directrices : x  = ± 
e
a
.
Focii : S ? (± a
 
e, 0). Length of, major axes = 2a and minor axes = 2b
Vertices :   A ? ? ? ( ? a, 0) & A ? (a, 0) .
Latus Rectum : = ? ?
2
2
e 1 a 2
a
b 2
? ?
2. Auxiliary Circle : x² + y² = a²
3. Parametric Representation : x = a cos ? & y = b sin ?
4. Position of a Point w.r.t. an Ellipse:
The point P(x
1,
 y
1
) lies outside, inside or on the ellipse according as;
 1
b
y
a
x
2
2
1
2
2
1
? ? > < or = 0.
5.
The line y = mx + c meets the ellipse 
2
2
2
2
b
y
a
x
? = 1 in two points real,
coincident or imaginary according as c² is < = or > a²m² + b².
6. Tangents:
Slope form: y = mx ± 
2 2 2
a m ? b
, Point form : 1
b
yy
a
xx
2
1
2
1
? ? ,
Parametric form:  1
b
ysin
a
xcos
?
?
?
?
7. Normals:
1
2
1
2
y
b y
x
a x
?
 = a² ? b²,  ax.
 
sec
 
? ? by
. 
cosec
 
? = (a² ? b²), y = mx ?
? ?
2 2 2
2 2
a b m
a b m
?
?
.
8. Director Circle: x² + y² = a² + b²
Page # 6
HYPERBOLA
1. Standard Equation:
Standard equation of the hyperbola is 1
2
b
2
y
2
a
2
x
? ? , where b
2
 = a
2 
(e
2
 ? 1).
Focii : S ? (± ae, 0) Directrices : x = ± 
a
e
Vertices : A ? ?(± a, 0)
Latus Rectum (
 
?
 
) : ? ?= 
a
b 2
2
 = 2a (e
2
 ?
 
1).
2. Conjugate Hyperbola :
1
b
y
a
x
2
2
2
2
? ?
   &  
1
b
y
a
x
2
2
2
2
? ? ?
 are conjugate hyperbolas of each.
3. Auxiliary Circle :  x
2
 + y
2
 = a
2
.
4. Parametric Representation : x = a sec ? ?& y = b tan ?
Page 5


Page # 4
CIRCLE
1. Intercepts made by Circle x
2 
+ y
2 
+ 2gx + 2fy + c = 0 on the Axes:
(a) 2 c g
2
? on x -axis (b) 2 c f
2
?  on y - aixs
2. Parametric Equations of a Circle:
x = h + r cos ? ;  y = k + r sin ?
3. Tangent :
(a) Slope form :  y = mx ± 
2
m 1 a ?
(b) Point form :  xx
1
 + yy
1
 = a
2
 or T = o
(c) Parametric form : x cos
 
? + y sin
 
? = a.
4. Pair of Tangents from a Point: SS
1
 = T².
5. Length of a Tangent : Length of tangent is
1
S
6. Director Circle: x
2
 + y
2
 = 2a
2
 for x
2
 + y
2
 = a
2
7. Chord of Contact: T = 0
1. Length of chord of contact =
2 2
L R
R L 2
?
2. Area of the triangle formed by the pair of the tangents & its chord of
contact = 
2 2
3
L R
L R
?
3. Tangent of the angle between the pair of tangents from (x
1
, y
1
)
= 
?
?
?
?
?
?
?
?
?
2 2
R L
L R 2
4. Equation of the circle circumscribing the triangle PT
1 
T
2
 is :
(x ? x
1
) (x + g) + (y ? y
1
) (y + f) = 0.
8. Condition of orthogonality of Two Circles:  2 g
1 
g
2 
+ 2 f
1 
f
2
 = c
1 
+ c
2
.
9. Radical Axis : S
1
 ? S
2
 = 0  i.e. 2
 
(g
1 
? g
2
) x + 2
 
(f
1 
? f
2
) y + (c
1 
? c
2
) = 0.
10. Family of Circles:  S
1
 + K S
2
 = 0, S + KL = 0.
Page # 5
PARABOLA
1. Equation of standard parabola :
y
2
 = 4ax, Vertex is (0, 0),  focus is (a, 0), Directrix is x + a = 0 and Axis is
y = 0. Length of the latus rectum = 4a, ends of the latus rectum are L(a, 2a)
& L’ (a, ? 2a).
2. Parametric Representation: x = at² & y = 2at
3. Tangents to the Parabola y² = 4ax:
1. Slope form y = mx +
m
a
 (m ? 0)2. Parametric form ty = x + at
2
3. Point form T = 0
4. Normals to the parabola y² = 4ax :
y ? y
1
 =
a 2
y
1
?
 (x ? x
1
) at (x
1,
 y
1
)  ; y = mx ? 2am ? am
3
 at (am
2
? ? ? 2am) ;
y + tx = 2at + at
3
 at (at
2
, 2at).
Page # 5
ELLIPSE
1. Standard Equation : 
2
2
2
2
b
y
a
x
? = 1, where a > b & b² = a² (1 ? e²).
Eccentricity:  e =
2
2
a
b
1 ? , (0 < e < 1),  Directrices : x  = ± 
e
a
.
Focii : S ? (± a
 
e, 0). Length of, major axes = 2a and minor axes = 2b
Vertices :   A ? ? ? ( ? a, 0) & A ? (a, 0) .
Latus Rectum : = ? ?
2
2
e 1 a 2
a
b 2
? ?
2. Auxiliary Circle : x² + y² = a²
3. Parametric Representation : x = a cos ? & y = b sin ?
4. Position of a Point w.r.t. an Ellipse:
The point P(x
1,
 y
1
) lies outside, inside or on the ellipse according as;
 1
b
y
a
x
2
2
1
2
2
1
? ? > < or = 0.
5.
The line y = mx + c meets the ellipse 
2
2
2
2
b
y
a
x
? = 1 in two points real,
coincident or imaginary according as c² is < = or > a²m² + b².
6. Tangents:
Slope form: y = mx ± 
2 2 2
a m ? b
, Point form : 1
b
yy
a
xx
2
1
2
1
? ? ,
Parametric form:  1
b
ysin
a
xcos
?
?
?
?
7. Normals:
1
2
1
2
y
b y
x
a x
?
 = a² ? b²,  ax.
 
sec
 
? ? by
. 
cosec
 
? = (a² ? b²), y = mx ?
? ?
2 2 2
2 2
a b m
a b m
?
?
.
8. Director Circle: x² + y² = a² + b²
Page # 6
HYPERBOLA
1. Standard Equation:
Standard equation of the hyperbola is 1
2
b
2
y
2
a
2
x
? ? , where b
2
 = a
2 
(e
2
 ? 1).
Focii : S ? (± ae, 0) Directrices : x = ± 
a
e
Vertices : A ? ?(± a, 0)
Latus Rectum (
 
?
 
) : ? ?= 
a
b 2
2
 = 2a (e
2
 ?
 
1).
2. Conjugate Hyperbola :
1
b
y
a
x
2
2
2
2
? ?
   &  
1
b
y
a
x
2
2
2
2
? ? ?
 are conjugate hyperbolas of each.
3. Auxiliary Circle :  x
2
 + y
2
 = a
2
.
4. Parametric Representation : x = a sec ? ?& y = b tan ?
Page # 7
5. A Point 'P' w.r.t. A Hyperbola :
S
1
 ? ? 1
b
y
a
x
2
2
1
2
2
1
? ? >, = or < 0 according as the point (x
1, 
y
1
) lies inside, on
or outside the curve.
6. Tangents :
(i) Slope Form  : y = m x
2 2 2
b m a ? ?
(ii) Point Form :  at the point (x
1,
 y
1
) is 1
b
y y
a
x x
2
1
2
1
? ? .
(iii) Parametric Form : 1
b
an t y
a
sec x
?
?
?
?
.
7. Normals :
(a) at the point P (x
1
, y
1
) is  
1
2
1
2
y
y b
x
x a
? = a
2
 + b
2
 = a
2 
e
2
.
(b) at the point P (a sec ?, b tan ?) is 
?
?
? tan
y b
sec
x a
 = a
2
 + b
2
 = a
2 
e
2
.
(c) Equation of normals in terms of its slope 'm' are y
= mx ? ?
? ?
2 2 2
2 2
m b a
m b a
?
?
.
8. Asymptotes : 
0
b
y
a
x
? ?
 and 0
b
y
a
x
? ? .
Pair of asymptotes :
0
b
y
a
x
2
2
2
2
? ?
.
9. Rectangular Or Equilateral Hyperbola : xy = c
2
,  eccentricity is
2
.
Vertices : (± c
, 
±c) ; Focii :
? ? c 2 , c 2 ? ?
. Directrices : x + y = ? c 2
Latus Rectum (l
 
) : ? = 2 2
 
c = T.A. = C.A.
Parametric equation x = ct, y = c/t, t ? R – {0}
Equation of the tangent at P
 
(x
1
, 
y
1
) is 
1 1
y
y
x
x
?
 = 2 & at P
 
(t) is 
t
x
+ t
 
y = 2
 
c.
Equation of the normal at P
 
(t) is x
 
t
3
 ? y
 
t = c
 
(t
4 
?
 
1).
Chord with a given middle point as (h, k) is kx + hy = 2hk.