# Plus One Maths Notes Chapter 11 Conic Sections

Kerala State Board New Syllabus Plus One Maths Notes Chapter 11 Conic Sections.

## Kerala Plus One Maths Notes Chapter 11 Conic Sections

I. Circle
A circle is the set of all points in a plane that are equidistant from a fixed point in the plane. The fixed point is the centre and the fixed distance is the radius.
Equation of a circle with centre origin and radius r is x2 + y2 = r2.

Equation of a circle with centre (h, k) and radius r is (x – h)2 + (y – k)2 = r2.

General form of the equation of a circle is x2 + y2 + 2gx + 2fy + c = 0 with centre (-g, -f) and radius $$\sqrt{g^{2}+f^{2}-c}$$.

II. Conic
A conic is the set of all points in a plane which moves so that the distance from a fixed point is in a constant ratio to its distance from a fixed-line. The fixed point is the focus and fixed line is directrix and the constant ratio is eccentricity, denoted by ‘e’.

III. Parabola (e = 1)

y2 = 4ax

Vertex: (0, 0)
Focus(S): (a, 0)
Length of Latusrectum: (LL’) = 4a
Equation of directrix (DD’) is x = -a

y2 = -4ax

Vertex: (0, 0)
Focus(S): (-a, 0)
Length of Latusrectum (LL’) = 4a
Equation of directrix (DD’) is x = a

x2 = 4ay

Vertex: (0, 0)
Focus(S): (0, a)
Length of Latusrectum (LL’) = 4a
Equation of directrix (DD’) is y = -a

x2 = -4ay

Vertex: (0, 0)
Focus(S): (0, -a)
Length of Latusrectum (LL’) = 4a
Equation of directrix (DD’) is y = a

IV. Ellipse (e < 1)

$$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$, a > b

1. Eccentricity, e = $$\frac{\sqrt{a^{2}-b^{2}}}{a}$$
(ae)2 = a2 – b2 ⇒ c2 = a2 – b2
2. b2 = a2(1 – e2)
3. Length of Latusrectum (LL’) = $$\frac{2 b^{2}}{a}$$
4. Focii, S(ae, 0) and S'(-ae, 0) or S(c, 0), S'(-c, 0)
5. Centre (0, 0)
6. Vertices A(a, 0) and A'(-a, 0)
7. Equation of directrix (DD’) is x = $$\frac{a}{e}$$ and x = $$-\frac{a}{e}$$
8. Length of major axis (AA’) = 2a
9. Length of minor axis'(BB’) = 2b

$$\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1$$, a > b

1. Eccentricity, e = $$\frac{\sqrt{a^{2}-b^{2}}}{a}$$
(ae)2 = a2 – b2 ⇒ c2 = a2 – b2
2. b2 = a2(1 – e2)
3. Length of Latus rectum (LL’) = $$\frac{2 b^{2}}{a}$$
4. Focii, S(0, ae) and S'(0, -ae) or S(0, c), S'(0, -c)
5. Centre (0, 0)
6. Vertices A(0, a) and A'(0, -a)
7. Equation of directrix (DD’) is y = $$\frac{a}{e}$$ and y = $$-\frac{a}{e}$$
8. Length of major axis (AA’) = 2a
9. Length of minor axis (BB’) = 2b

V. Hyperbola (e > 1)

$$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$$

1. Eccentricity, e = $$\frac{\sqrt{a^{2}+b^{2}}}{a}$$
(ae)2 = a2 + b2 ⇒ c2 = a2 + b2
2. b2 = a2(e2 – 1)
3. Length of Latus rectum (LL’) = $$\frac{2 b^{2}}{a}$$
4. Focii, S(ae, 0) and S'(-ae, 0) or S(c, 0), S'(-c, 0)
5. Centre (0, 0)
6. Vertices A(a, 0) and A'(-a, 0)
7. Equation of directrix (DD’) is x = $$\frac{a}{e}$$ and x = $$-\frac{a}{e}$$

$$\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1$$

1. Eccentricity, e = $$\frac{\sqrt{a^{2}+b^{2}}}{a}$$
(ae)2 = a2 + b2 ⇒ c2 = a2 + b2
2. b2 = a2(e2 – 1)
3. Length of Latus rectum (LL’) = $$\frac{2 b^{2}}{a}$$
4. Focii, S(0, ae) and S'(0, -ae) or S(0, c), S'(0, -c)
5. Centre (0, 0)
6. Vertices A(0, a) anti A'(0, -a)
7. Equation of directrix (DD’) is y = $$\frac{a}{e}$$ and y = $$-\frac{a}{e}$$