CLASS XII CONIC SECTION
CLASS XII – CONIC SECTION
(1) CIRCLE
(i) Circle: Locus of a Point 5. Standard form of circle which has a constant
Distance from a fined Point.
ii) equation of circle which have centre (h, k) and radius 'r'
(x-h)² + (y-k)² = r²
Ex: find equation of circle having centre (2,3) and radius 5 unit
Sol. (x-2)² + (y-3)² = (5)²
3. Equation of circle having co-ordinate of diameter (x,y₁),(x₂,y₂)
(x-x₁) (x-x₂) + (y-y₁) (y-y₂) = 0
4. Equation of circle whose centre (0,0) and radius r
x² + y² = r²
5. Standard form of circle
x² + y² + 2gx + 2fy + C = 0
(i) centre (-g, -f)
(ii) radius r = √(g² + f² - c)
Note: to find centre and radius coefficients of x and y should be unit. If
not unit then, divide the equation by coeff. Of x² or y²
6. r = √(a²/4 + b²/4)
7. intercept on x-axis
= 2√(g² - c)
8. intercept on y-axis
= 2√(f² - c)
Parabola
If PS = MP then locus
of P(x,y) form a Parabola.
Type 1: Standard form
y²=4ax
i. focus (a,0)
ii. vertex (0,0)
iii. Eq. of directrix x= -a
iv. length of latus rectum = 2a
v. Equation of axis y = 0
Type 2: Standard form.
y² = -4ax
1. focus (-a,0)
2 vertex (0,0)
3 Eq. of directrix x = a
Type3: Standard form.
x²=4ay
1) focus (0,a)
2) Vertex (0,0)
3. Eq. of directrix y = -a
4. length of Latus rectum = 4a
Type4: Standard form.
x²=-4ay
1) focus (0,-a)
2) vertex (0,0)
3) Eq. of directrix y=a.
4) length of L.R. = 4a
(2) ELLIPSE
✤ Case: Standard form
x²/a² + y²/b² = 1 a > b
1. Eccentricity e = √(1-b²/a²), e<1
2. PS1/PM < 1
3. foci (±ae, 0)
4. Vertex (±a, 0)
5. equation of directrix x = ±a/e
6. Length of Latus rectum = 2b²/a
7. Length of major axis = 2a
8. Length of minor axis = 2b
9. Equation of major axis y = 0
10. Equation of minor axis x = 0
✤ Case2: b > a
1. e = √1-a²/b²
2. foci (0, ±be)
3. vertex (0,±b)
4. PS/PM < 1
5. equation of directrix- m, y=±b/e
6. Length of latus rectum = 2a²/b
7. Length of major axis = 2b
8. Length of minor axis = 2a
9. Eq. of major axis x=0
10. minor axis y = 0
Note: c = √a²-b² when a>b
c= √b²-a², when b>a
(3) HYPERBOLA
CASE 1 :
X2 / a2 – y2 / b2 = 1
1. e = √1+b²/a², e = 1/e
2. foci (±ae, 0)
3. vertex (±a, 0)
4. equation of directrix, x = ±a/e
5. Length of latus rectum = 2b²/a
6. Length of transverse axis = 2a
7. Length of Major axis = 2b
(Conjugate)
8. Eq. of transverse axis, y = 0
9. Eq. of Conjugate axis, x = 0
Ex. x²/16 - y²/7 = 1
CASE 2 :
- x2 / a2 + y2 / b2 = 1
1. e = √(1 + a²/d²)
2. foci (0, ±ke)
3. vertex (0, ±k)
4. equation of directrix, y = ± k/e
5. equation of Transverse axis, x = 0
6. equation of Conjugate axis, x = 0
7. length of transverse axis = 2a
8. equation of Conjugate axis = 2b
9. Length of latus rectum = 2a²/b
Ex. -x²/9 + y²/32 = 1
