Learning these TS Inter 2nd Year Maths 2B Formulas Chapter 3 Parabola will help students to solve mathematical problems quickly.

## TS Inter 2nd Year Maths 2B Parabola Formulas

→ The locus of a point moving on a plane such that its distance from a fixed point and a fixed straight line in the plane are in a constant ratio ‘e’ is called a conic and e is called the eccentricity of the conic. Fixed point is called focus and the fixed straight line is called the directrix. If e = 1, the conic is called a Parabola.

→ Equation of a parabola in the standard form is y2 = 4ax (a > 0) with focus (a, 0), axis y = 0, equation of directrix is x + a = 0 and length of the latus rectum is 4a.

→ Various different forms of parabola (general)

Table: 1

(i) Form: y^{2} = 4ax

Vertex: (0, 0)

Focus: (a, 0)

Directrix: x + a = 0

Axis: y = 0

(ii) Form: x^{2} = 4ay

Vertex: (0, 0)

Focus: (0, a)

Directrix: y + a = 0

Axis: x = 0

(iii) Form: y^{2} = -4ax

Vertex: (0, 0)

Focus: (-a, 0)

Directrix: x – a = 0

Axis: y = 0

(iv) Form: x^{2} = -4ay

Vertex: (0, 0)

Focus: (0, -a)

Directrix: y – a = 0

Axis: x = 0

Table: 2

(i) Form:(y – k)^{2} = 4a(x – h)

Vertex: (h, k)

Focns: (h + a, k)

Axis: y = k

EquaffonotDlrectrlx: x – h + a = 0

(ii) Form: (y – k)^{2} = -4a(x – h)

Vertex: (h, k)

Focus: (h – a, k)

Axis: y – k = 0

Equation of Directrix: x – h – a = 0

(iii) Form: (x – h)^{2} = -4a(y – k)

Vertex: (h, k)

Focus: (h, k – a)

Axis: x – h = 0

Equation of Directrix : y – k -a = 0

(iv) Form : (x – h)^{2} = 4a(y – k)

Vertex: (h, k)

Focus: (h, a + k)

Axis: x – h = 0

Equation of Directrix: y – k + a = 0

(v) Form: (x – α)^{2} + (y – β)^{2} = \(\frac{(l x+m y+n)^2}{l^2+m^2}\)

Vertex : Point A

Focus: (α, β)

Axis: m(x – α) – l(y – β) = 0

Equation of Directrix: lx + my + n = 0

→ The equation of parabola when its axis is parallel to X – axis is x = ly^{2} + my + n and when axis is parallel to Y- axis is y = lx^{2} + mx + n.

→ Focal distance of a point (x_{1}, y_{1}) on the parabola y^{2} = 4ax is x_{1} + a.

→ Parametric equations of the parabola are x = at^{2}, y = 2at.

→ P(x_{1}, y_{1}) lies outside, on or inside the parabola y^{2} = 4ax according as S_{11} \(\frac{\geq}{<}\) 0.

→ y = mx + c (m * 0) is a tangent to the parabola y^{2} = 4ax when c = a/m.

→ y = mx + a/m (m ≠ 0) is always a tangent to the parabola y^{2} = 4ax and point of contact is \(\left(\frac{a}{m^2}, \frac{2 a}{m}\right)\)

→ Equation of the tangent at the point (x_{1}, y_{1}) on the parabola y^{2} = 4ax is yy_{1} = 2a (x + x_{1}) or S_{1} = 0. .

→ Equation of the normal at the point (x_{1}, y_{1}) on the parabola S = 0 is y – y_{1} = \(-\frac{y_1}{2 a}\) (x – x_{1}).

→ Equation of tangent at a point ‘t’ i.e„ (at^{2}, 2at) on y^{2} = 4ax is x = yt + at^{2}.

→ Equation of normal at a point ‘t’ on y^{2} = 4ax is y + xt = 2at + at^{3}.