MATHEMATICS
STRAIGHT LINE
q
2 2
1. Distance Formula: d = (x1 − x2 ) + (y1 − y2 ) .
mx2 ±nx1 my2 ±ny1
2. Section Formula : x = m±n ; y = m±n .
3. Centroid, Incentre & Excentre:
x1 + x2 + x3 y1 + y2 + y3 ax1 + bx2 + cx3 ay1 + by2 + cy3
Centroid G , , Incentre I ,
3 3 a+b+c a+b+c
−ax1 + bx2 + cx3 −ay1 + by2 + cy3
Excentre I1 ,
−a + b + c −a + b + c
4. Area of a Triangle:
x1 y1 1
1
△ABC = 2
x2 y2 1
x3 y3 1
5. Slope Formula:
y1 −y2
(i) Line Joining two points (x1 y1 ) & (x2 y2 ) , m = x1 −x2
x1 y1 1
6. Condition of collinearity of three points: x2 y2 1 =0
x3 y3 1
m1 −m2
7. Angle between two straight lines : tan θ = 1+m1 m2 .
8. Two Lines : ax + by + c = 0 and a′ x + b′ y + c′ = 0 two lines
a b c
9. parallel if a′ = b′ ̸= c′ .
10. Distance between two parallel lines = √c1 −c2 .
a2 +b2
ax√
1 +by1 +c
11. Distance between point and line = a2 +b2
.
1
, y−y1
12. Reflection of a point about a line: x−x1
a = b = −2 ax1a+by 1 +c
2 +b2
x−x1 y−y1
13. Foot of the perpendicular from a point on the line is a = b =
− ax1a+by 1 +c
2 +b2
′ ′ ′
ax+by+c
14. Bisectors of the angles between two lines: √
a2 +b2
= ± a√x+b y+c
a′2 +b′2
15. Condition of Concurrency :of three straight lines a1 x + by + c1 = 0, i =
a1 b1 c1
1, 2, 3 is a2 b2 c2 = 0.
a3 b3 c3
16. A Pair of straight lines through origin: ax2 + 2hxy + by 2 = 0
If √θ is the acute angle between the pair of straight lines, then tan θ =
2 h2 −ab
a+b .
CIRCLE
1. Intercepts
p made by Circle x2 + y 2 + 2gx + 2f y + c = 0 on the Axes:
2
(a) 2 pg − c on x-axis
(b) 2 f 2 − c on y - aixs
2. Parametric Equations of a Circle: x = h + r cos θ; y = k + r sin θ
3. Tangent :
√
(a) Slope form : y = mx ± a 1 + m2
(b) Point form : x1 + y1 = a2 or T = 0
(c) Parametric form : x cos α + y sin α = a.
4. Pair of Tangents from a Point: SS1 = T2 .
√
5. Length of a Tangent : Length of tangent is S1
6. Director Circle: x2 + y 2 = 2a2 for x2 + y 2 = a2
7. Chord of Contact: T = 0
8. Length of chord of contact = √ 2LR
R2 +L2
9. Area of the triangle formed by the pair of the tangents & its chord of
3
contact = RRL
2 +L2
10. Tangent of the angle between the pair of tangents from (x1 , y1 ) = L2RL
2 −R2
11. Equation of the circle circumscribing the triangle P T1 T2 is : (x − x1 ) (x +
g) + (y − y1 ) (y + f ) = 0.
2
,12. Condition of orthogonality of Two Circles: 2g1 g2 + 2f1 f2 = c1 + c2 .
13. Radical Axis : S1 − S2 = 0 i.e. 2 (g1 − g2 ) x + 2 (f1 − f2 ) y + (c1 − c2 ) = 0.
14. Family of Circles: S1 + KS2 = 0, S + KL = 0.
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 y 2 = 4ax :
a
4. Slope form y = mx + m (m ̸= 0)
5. Parametric form ty = x + at2
6. Point form T = 0
7. Normals to the parabola y 2 = 4ax :
y1
y − y1 = − 2a (x − x1 ) at (x1 , y1 ) ;
y = mx − 2am − am3 at am2, −2am ;
y + tx = 2at + at3 at at2 , 2at .
ELLIPSE
2 2
1. Standard Equation : xa2 + yb2 = 1, where a > b & b2 = a2 1 − e2 .
q
2
Eccentricity: e = 1 − ba2 , (0 < e < 1), Directrices : x = ± ae
Focii : S ≡ (±ae, 0). Length of, major axes = 2a and minor axes = 2 b
Vertices : A′ ≡ (−a, 0)&A ≡ (a, 0)
2
Latus Rectum : = 2ba = 2a 1 − e2
2. Auxiliary Circle : x2 + y 2 = a2
3. Parametric Representation : x = a cos θ&y = b sin θ
4. Position of a Point w.r.t. an Ellipse: The point P (x1 , y1 ) lies outside,
x2 y2
inside or on the ellipse according as ; a21 + b21 − 1 >< or = 0.
2 2
5. Line and an Ellipse: The line y = mx + c meets the ellipse xa2 + yb2 = 1
in two points real, coincident or imaginary according as c2 is <= or >
a2 m2 + b2 .
3
, √
6. Tangents: Slope form: y = mx ± a2 m2 + b2 , Point form : xx1
a2 + yy
b2 = 1,
1
x cos θ y sin θ
Parametric form: a + b =1
2 b2 y
7. Normals: ax1x −
y1 = a2 − b2 , ax · sec θ − by cosec θ = a2 − b2 , y =
(a2 −b2 )m
mx − √a2 +b2 m2 .
8. Director Circle: x2 + y 2 = a2 + b2
HYPERBOLA
x2 y2
1. Standard Equation: Standard
equation of the hyperbola is a2 − b2 = 1,
where b2 = a2 e2 − 1 .
Focii : S = (± ae, 0) Directrices : x = ± ae
Vertices: A = (±a, 0)
2
Latus Rectum (ℓ) : ℓ = 2 ab = 2a e2 − 1 ).
x2 y2 x2 y2
2. Conjugate Hyperbola : a2 − b2 = 1 &− a2 + b2 = 1 are conjugate
hyperbolas of each.
3. Auxiliary Circle : x2 + y 2 = a2 .
4. Parametric Representation : x = a sec θ&y = b tan θ
2 2
5. Position of A Point ’P’ w.r.t. A Hyperbola : S1 ≡ xa12 − yb1 2 − 1 >, = or
< 0 according as the point (x1, y1 ) lies inside, on or outside the curve.
6. Tangents : √
(i) Slope Form : y = m × ± a2 m2 − b2
xx1 yy1
(ii) Point Form : at the point (x1, y1 ) is a2 − b2 = 1.
x sec θ y tan θ
(iii) Parametric Form : a − b = 1.
a2 x b2 y 2 2 2 2
Normals (a) at the point P (x1 , y1 ) is x1 + y1 = a + b = a e . (b) at
ax by 2 2 2 2
the point P (a sec θ, b tan θ) is sec θ + tan θ = a + b = a e . (c) Equation
(a2 +b2 )m
of normals in terms of its slope ’ m ’ are y = mx ± √a2 −b2 m2 .
2
x2
7. Asymptotes : x
a + by = 0 and x
a − by = 0. Pair of asymptotes:
− yb2 = 0. a2
√
8. Rectangular Or Equilateral Hyperbola : xy = c2 , eccentricity is 2.
√ √ √
(± 2c ± 2c). Directrices : x + y = ± 2c
Vertices : (±c ± c); Focii : √
Latus Rectum ( l ) : ℓ = 2 2c = T.A. = C.A.
Parametric equation x = ct, y = c/t, t ∈ R − {0}
Equation of the tangent at P (x1 y1 ) is xx1 + yy1 = 2& at P (t) is xt + ty = 2c.
Equation of the normal at P (t) is xt3 − yt = c t4 − 1 .
Chord with a given middle point as (h, k) is kx + hy = 2hk.
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STRAIGHT LINE
q
2 2
1. Distance Formula: d = (x1 − x2 ) + (y1 − y2 ) .
mx2 ±nx1 my2 ±ny1
2. Section Formula : x = m±n ; y = m±n .
3. Centroid, Incentre & Excentre:
x1 + x2 + x3 y1 + y2 + y3 ax1 + bx2 + cx3 ay1 + by2 + cy3
Centroid G , , Incentre I ,
3 3 a+b+c a+b+c
−ax1 + bx2 + cx3 −ay1 + by2 + cy3
Excentre I1 ,
−a + b + c −a + b + c
4. Area of a Triangle:
x1 y1 1
1
△ABC = 2
x2 y2 1
x3 y3 1
5. Slope Formula:
y1 −y2
(i) Line Joining two points (x1 y1 ) & (x2 y2 ) , m = x1 −x2
x1 y1 1
6. Condition of collinearity of three points: x2 y2 1 =0
x3 y3 1
m1 −m2
7. Angle between two straight lines : tan θ = 1+m1 m2 .
8. Two Lines : ax + by + c = 0 and a′ x + b′ y + c′ = 0 two lines
a b c
9. parallel if a′ = b′ ̸= c′ .
10. Distance between two parallel lines = √c1 −c2 .
a2 +b2
ax√
1 +by1 +c
11. Distance between point and line = a2 +b2
.
1
, y−y1
12. Reflection of a point about a line: x−x1
a = b = −2 ax1a+by 1 +c
2 +b2
x−x1 y−y1
13. Foot of the perpendicular from a point on the line is a = b =
− ax1a+by 1 +c
2 +b2
′ ′ ′
ax+by+c
14. Bisectors of the angles between two lines: √
a2 +b2
= ± a√x+b y+c
a′2 +b′2
15. Condition of Concurrency :of three straight lines a1 x + by + c1 = 0, i =
a1 b1 c1
1, 2, 3 is a2 b2 c2 = 0.
a3 b3 c3
16. A Pair of straight lines through origin: ax2 + 2hxy + by 2 = 0
If √θ is the acute angle between the pair of straight lines, then tan θ =
2 h2 −ab
a+b .
CIRCLE
1. Intercepts
p made by Circle x2 + y 2 + 2gx + 2f y + c = 0 on the Axes:
2
(a) 2 pg − c on x-axis
(b) 2 f 2 − c on y - aixs
2. Parametric Equations of a Circle: x = h + r cos θ; y = k + r sin θ
3. Tangent :
√
(a) Slope form : y = mx ± a 1 + m2
(b) Point form : x1 + y1 = a2 or T = 0
(c) Parametric form : x cos α + y sin α = a.
4. Pair of Tangents from a Point: SS1 = T2 .
√
5. Length of a Tangent : Length of tangent is S1
6. Director Circle: x2 + y 2 = 2a2 for x2 + y 2 = a2
7. Chord of Contact: T = 0
8. Length of chord of contact = √ 2LR
R2 +L2
9. Area of the triangle formed by the pair of the tangents & its chord of
3
contact = RRL
2 +L2
10. Tangent of the angle between the pair of tangents from (x1 , y1 ) = L2RL
2 −R2
11. Equation of the circle circumscribing the triangle P T1 T2 is : (x − x1 ) (x +
g) + (y − y1 ) (y + f ) = 0.
2
,12. Condition of orthogonality of Two Circles: 2g1 g2 + 2f1 f2 = c1 + c2 .
13. Radical Axis : S1 − S2 = 0 i.e. 2 (g1 − g2 ) x + 2 (f1 − f2 ) y + (c1 − c2 ) = 0.
14. Family of Circles: S1 + KS2 = 0, S + KL = 0.
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 y 2 = 4ax :
a
4. Slope form y = mx + m (m ̸= 0)
5. Parametric form ty = x + at2
6. Point form T = 0
7. Normals to the parabola y 2 = 4ax :
y1
y − y1 = − 2a (x − x1 ) at (x1 , y1 ) ;
y = mx − 2am − am3 at am2, −2am ;
y + tx = 2at + at3 at at2 , 2at .
ELLIPSE
2 2
1. Standard Equation : xa2 + yb2 = 1, where a > b & b2 = a2 1 − e2 .
q
2
Eccentricity: e = 1 − ba2 , (0 < e < 1), Directrices : x = ± ae
Focii : S ≡ (±ae, 0). Length of, major axes = 2a and minor axes = 2 b
Vertices : A′ ≡ (−a, 0)&A ≡ (a, 0)
2
Latus Rectum : = 2ba = 2a 1 − e2
2. Auxiliary Circle : x2 + y 2 = a2
3. Parametric Representation : x = a cos θ&y = b sin θ
4. Position of a Point w.r.t. an Ellipse: The point P (x1 , y1 ) lies outside,
x2 y2
inside or on the ellipse according as ; a21 + b21 − 1 >< or = 0.
2 2
5. Line and an Ellipse: The line y = mx + c meets the ellipse xa2 + yb2 = 1
in two points real, coincident or imaginary according as c2 is <= or >
a2 m2 + b2 .
3
, √
6. Tangents: Slope form: y = mx ± a2 m2 + b2 , Point form : xx1
a2 + yy
b2 = 1,
1
x cos θ y sin θ
Parametric form: a + b =1
2 b2 y
7. Normals: ax1x −
y1 = a2 − b2 , ax · sec θ − by cosec θ = a2 − b2 , y =
(a2 −b2 )m
mx − √a2 +b2 m2 .
8. Director Circle: x2 + y 2 = a2 + b2
HYPERBOLA
x2 y2
1. Standard Equation: Standard
equation of the hyperbola is a2 − b2 = 1,
where b2 = a2 e2 − 1 .
Focii : S = (± ae, 0) Directrices : x = ± ae
Vertices: A = (±a, 0)
2
Latus Rectum (ℓ) : ℓ = 2 ab = 2a e2 − 1 ).
x2 y2 x2 y2
2. Conjugate Hyperbola : a2 − b2 = 1 &− a2 + b2 = 1 are conjugate
hyperbolas of each.
3. Auxiliary Circle : x2 + y 2 = a2 .
4. Parametric Representation : x = a sec θ&y = b tan θ
2 2
5. Position of A Point ’P’ w.r.t. A Hyperbola : S1 ≡ xa12 − yb1 2 − 1 >, = or
< 0 according as the point (x1, y1 ) lies inside, on or outside the curve.
6. Tangents : √
(i) Slope Form : y = m × ± a2 m2 − b2
xx1 yy1
(ii) Point Form : at the point (x1, y1 ) is a2 − b2 = 1.
x sec θ y tan θ
(iii) Parametric Form : a − b = 1.
a2 x b2 y 2 2 2 2
Normals (a) at the point P (x1 , y1 ) is x1 + y1 = a + b = a e . (b) at
ax by 2 2 2 2
the point P (a sec θ, b tan θ) is sec θ + tan θ = a + b = a e . (c) Equation
(a2 +b2 )m
of normals in terms of its slope ’ m ’ are y = mx ± √a2 −b2 m2 .
2
x2
7. Asymptotes : x
a + by = 0 and x
a − by = 0. Pair of asymptotes:
− yb2 = 0. a2
√
8. Rectangular Or Equilateral Hyperbola : xy = c2 , eccentricity is 2.
√ √ √
(± 2c ± 2c). Directrices : x + y = ± 2c
Vertices : (±c ± c); Focii : √
Latus Rectum ( l ) : ℓ = 2 2c = T.A. = C.A.
Parametric equation x = ct, y = c/t, t ∈ R − {0}
Equation of the tangent at P (x1 y1 ) is xx1 + yy1 = 2& at P (t) is xt + ty = 2c.
Equation of the normal at P (t) is xt3 − yt = c t4 − 1 .
Chord with a given middle point as (h, k) is kx + hy = 2hk.
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