Parabola Formulas

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1. Definition of conic section:

Its locus of a point ratio of whose distances from a fixed point and a fixed line is constant
This Ratio is called “eccentricity” and denoted by “e”.
If e = 1, conic section is parabola
e > 1, conic section is Hyperbola
e < 1, conic section is Ellipse
fixed line is called “Directrix”
Fixed point is called “Focus”

A₁A₂ = Latus rectum = 4 (OS) = 2(SS’)

2. Definition

A parabola is the locus of a point which moves in such a way that its distance from a fixed point is equal to its perpendicular distance from a fixed straight line.

• Focus: The fixed point is called the focus of the Parabola.
• Directrix: The fixed line is called the directrix of the Parabola.
• Eccentricity: For the parabola, e = 1.
  If e > 1 ⇒ Hyperbola, e = 0 ⇒ Circle, e < 1 ⇒ Ellipse

3. Double Ordinate

The chord which is perpendicular to the axis of parabola or parallel to directrix is called double ordinate of the parabola.

4. Length of Latus Rectum

The length of the Latus Rectum = 2 × perpendicular distance of focus from the directrix.

5. Parameters of the Parabola y² = 4ax

• Vertex O ⇒ (0, 0)
• Focus S ⇒ (a, 0)
• Directrix ⇒ x + a = 0
• Axis ⇒ y = 0 or x-axis
• Equation of Latus Rectum ⇒ x = a
• Length of Latus Rectum ⇒ 4a
• Ends of Latus Rectum ⇒ (a, 2a), (a, – 2a)
• The focal distance ⇒ sum of abscissa of the point and distance between vertex and Latus Rectum
• If length of any double ordinate of parabola y² = 4ax is 2l then coordinates of end points of this double ordinate are
  \(\left(\frac{\ell^{2}}{4 \mathrm{a}}, \ell\right) \text { and }\left(\frac{\ell^{2}}{4 \mathrm{a}},-\ell\right)\)

6. Parametric equation of Parabola

y² = 4ax are x = at², y = 2at and for parabola x² = 4ay is x = 2at, y = at²

7. Equation of chord joining any two point of a parabola

If the points are (at₁², 2at₁) and (at₂², 2at₂) then the equation of chord is (t₁ + t₂)y = 2x + 2at₁t₂

• If ‘t₁‘ and ‘t₂‘ are the parameters of the ends of a focal chord of the parabola y² = 4ax, then t₁t₂ = – 1
• If one end of focal chord of parabola is (at², 2at) , then other end will be (a/t², – 2at) and length of focal chord = a\(\left(t+\frac{1}{t}\right)^{2}\).
• The length of the chord joining two points ‘t₁‘ and ‘t₂‘ on the parabola y² = 4ax is a (t₁ – t₂) \(\sqrt{\left(t_{1}+t_{2}\right)^{2}+4}\)

8. Length of intercept

The length of intercept made by line y = mx + c between the parabola y² = 4ax is AB = \(\frac{4}{m^{2}} \sqrt{a\left(1+m^{2}\right)(a-m c)}\)

9. Condition of Tangency

• The line y = mx + c touches a parabola y² = 4ax then c = \(\frac{a}{m}\)
• The line y = mx + c touches a parabola x² = 4ay if c = – am²

10. Equation of Tangent

(i) Point Form:
The equation of tangent to the parabola y² = 4ax at the point (x₁, y₁) is yy₁ = 2a(x + x₁) or T = 0

(ii) Parametric Form:
The equation of the tangent to the parabola at t. i.e. (at², 2at) is ty = x + at²

(iii) Slope Form:
The equation of the tangent of the parabola y² = 4ax is y = mx + \(\frac{a}{m}\)

11. Equation of normal

(i) Point Form:
The equation to the normal at the point (x₁, y₁) of the parabola y² = 4axis given by y – y₁ = \(\frac{-y_{1}}{2 a}\)(x – x₁)

(ii) Parametric Form:
The equation to the normal at the point (at², 2at) is y + tx = 2at + at³

(iii) Slope Form:
Equation of normal in terms of slope m is y = mx – 2am – am³.

(iv) The foot of the normal is (am², -2am)

12. Condition of normal

The line y = mx + c is a normal to the parabola

• y² = 4ax if c = -2am – am³
• x² = 4ay if c = 2a + \(\frac{\mathrm{a}}{\mathrm{m}^{2}}\)

13. Normal Chord

If the normal at ’t₁‘ meets the parabola y² = 4ax again at the point ‘t₂‘ then this is called as normal chord. Again for normal chord, t₂ = -t₁ – \(\frac{2}{t_{1}}\)

14. Length of Normal chord is given by

a(t₁ – t₂) \(\sqrt{\left(t_{1}+t_{2}\right)^{2}+4}=\frac{4 a\left(t_{1}^{2}+1\right)^{3 / 2}}{t_{1}^{2}}\)

15. If two normal drawn at point ‘t₁‘ and ‘t₂‘ meet on the parabola then t₁t₂ = 2

16. Pair of Tangents: SS₁ = T²

17. Chord of contact

(i) The equation of chord of contact of tangents drawn from a point (x₁, y₁) to the parabola y² = 4ax is yy₁ = 2a (x + x₁).

(ii) Lengths of the chord of contact is \(\frac{1}{a} \sqrt{\left(y_{1}^{2}-4 a x\right)\left(y_{1}^{2}+4 a^{2}\right)}\)

(iii) Area of triangle formed by tangents drawn from (x₁, y₁) and their chord of contact is \(\frac{1}{2 a}\left(y_{1}^{2}-4 a x_{1}\right)^{3 / 2}\)

18. Equation of Polar

Equation of polar of the point (x₁, y₁) with respect to parabola y² = 4ax is yy₁ = 2a(x + x₁) or T = 0

19. Coordinates of Pole

The pole of the line £x + my + n =0 with respect to the parabola
y² = 4ax is \(\left(\frac{\mathrm{n}}{\ell}, \frac{-2 \mathrm{am}}{\ell}\right)\)

20. Diameter of the Parabola

The equation of a system of parallel chord y = mx + c with respect to the parabola y² = 4ax is given by y = \(\frac{2 a}{m}\)

21. The semi latus rectum of a parabola is the H.M. between the segments of any focal chord of a parabola i.e. if PQR is a focal chord, then 2a = \(\frac{2 \mathrm{PQ} \cdot \mathrm{QR}}{\mathrm{PQ}+\mathrm{QR}}\)

22. The area of triangle formed inside the parabola y² = 4ax is \(\frac{1}{8 a}\)(y₁ – y₂)(y₂ – y₃) (y₃ – y₁) where y₁, y₂, y₃ are ordinate of vertices of the triangle.

23. The area of triangle formed by three points on a parabola is twice the area of the triangle formed by the tangents at these points.

Standard Parabola’s ⇒

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