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 x^{2} + y^{2} = r^{2}.

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

General form of the equation of a circle is x^{2} + y^{2} + 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)

y^{2} = 4ax

Vertex: (0, 0)

Focus(S): (a, 0)

Length of Latusrectum: (LL’) = 4a

Equation of directrix (DD’) is x = -a

y^{2} = -4ax

Vertex: (0, 0)

Focus(S): (-a, 0)

Length of Latusrectum (LL’) = 4a

Equation of directrix (DD’) is x = a

x^{2} = 4ay

Vertex: (0, 0)

Focus(S): (0, a)

Length of Latusrectum (LL’) = 4a

Equation of directrix (DD’) is y = -a

x^{2} = -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} = a^{2} – b^{2} ⇒ c^{2} = a^{2} – b^{2}

2. b^{2} = a^{2}(1 – e^{2})

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} = a^{2} – b^{2} ⇒ c^{2} = a^{2} – b^{2}

2. b^{2} = a^{2}(1 – e^{2})

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} = a^{2} + b^{2} ⇒ c^{2} = a^{2} + b^{2}

2. b^{2} = a^{2}(e^{2} – 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} = a^{2} + b^{2} ⇒ c^{2} = a^{2} + b^{2}

2. b^{2} = a^{2}(e^{2} – 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 = \(-\frac{a}{e}\)