Important Formulas: Parabola | Mathematics (Maths) for JEE Main & Advanced PDF Download

Conic Sections:

A conic section, or conic is the locus of a point which moves in a plane so that its distance from a fixedpoint is in a constant ratio to its perpendicular distance from a fixed straight line.

• The fixed point is called the Focus.
• The fixed straight line is called the Directrix.
• The constant ratio is called the Eccentricity denoted by e.
• The line passing through the focus & perpendicular to the directrix is called the Axis.
• A point of intersection of a conic with its axis is called a Vertex.

General Equation Of A Conic : Focal Directrix Property :
The general equation of a conic with focus (p, q) & directrix lx + my + n = 0 is :

(l² + m²) [(x − p)² + (y − q)²] = e²(lx + my + n)² ≡ ax² + 2hxy + by² + 2gx + 2fy + c = 0
Distinguishing Between The Conic :
The nature of the conic section depends upon the position of the focus S w.r.t. the directrix & also upon the value of the eccentricity e. Two different cases arise.
Case (I) : When The Focus Lies On The Directrix.
In this case D ≡ abc + 2fgh − af² − bg² − ch² = 0 & the general equation of a conic represents a pair of straight lines if :
e > 1 the lines will be real & distinct intersecting at S.
e = 1 the lines will coincident.
e < 1 the lines will be imaginary.
Case (II) : When The Focus Does Not Lie On Directrix.

Parabola

Definition :
A parabola is the locus of a point which moves in a plane, such that its distance from a fixed point (focus) is equal to its perpendicular distance from a fixed straight line (directrix).
Standard equation of a parabola is y2 = 4ax. For this parabola :
(i) Vertex is (0, 0)
(ii) focus is (a, 0)
(iii) Axis is y = 0
(iv) Directrix is x + a = 0
Focal Distance:
The distance of a point on the parabola from the focus is called the Focal Distance Of The Point.
Focal Chord :
A chord of the parabola, which passes through the focus is called a Focal Chord.
Double Ordinate :
A chord of the parabola perpendicular to the axis of the symmetry is called a Double Ordinate.
Latus Rectum:
A double ordinate passing through the focus or a focal chord perpendicular to the axis of parabola is called the Latus Rectum. For y² = 4ax.
Length of the latus rectum = 4a.
ends of the latus rectum are L(a, 2a) & L' (a, − 2a).
Note that:
(i) Perpendicular distance from focus on directrix = half the latus rectum.
(ii) Vertex is middle point of the focus & the point of intersection of directrix & axis.
(iii) Two parabolas are laid to be equal if they have the same latus rectum.
Four standard forms of the parabola are y²= 4ax ; y² = − 4ax ; x² = 4ay ; x² = − 4ay

Position Of A Point Relative To A Parabola :
The point (x₁ y₁) lies outside, on or inside the parabola y² = 4ax according as the expression y₁² − 4ax₁ is positive, zero or negative.

Line & A Parabola :
The line y = mx + c meets the parabola y² = 4ax in two points real, coincident or imaginary according as  c m  condition of tangency is,

7. Length of the chord intercepted by the parabola on the line y = m x + c is :Note: length of the focal chord making an angle a with the x− axis is 4aCosec² a.

Parametric Representation :
The simplest & the best form of representing the co−ordinates of a point on the parabola is (at², 2at).
The equations x = at² & y = 2at together represents the parabola y² = 4ax, t being the parameter. The equation of a chord joining t₁ & t₂ is 2x − (t₁ + t₂) y + 2 at₁ t₂ = 0.
Note: If chord joining t₁, t₂ & t₃, t₄ pass a through point (c, 0) on axis, then t₁t₂ = t₃t₄ = − c/a.

Tangents to the Parabola y² = 4ax:
(i) y y₁ = 2 a (x + x₁) at the point (x₁, y₁) ;
(ii)
(iii) t y = x + a t² at (at², 2at).
Note : Point of intersection of the tangents at the point t₁ & t₂ is [ at₁ t₂, a(t₁ + t₂) ].

Normals to the Parabola y² = 4ax :
(i) y − y₁ = (x − x₁) at (x₁, y₁) ;
(ii) y = mx − 2am − am³ at (am², − 2am)
(iii) y + tx = 2at + at³ at (at², 2at).
Note : Point of intersection of normals at t₁ & t₂ are, a (t₁² + t₂² + t₁t² + 2) ; − a t₁ t₂ (t₁ + t₂).

Three Very Important Results :
(a) If t₁ & t₂ are the ends of a focal chord of the parabola y² = 4ax then t₁t₂ = −1. Hence the co-ordinates at the extremities of a focal chord can be taken as (at², 2at) &(b) If the normals to the parabola y² = 4ax at the point t₁, meets the parabola again at the point t₂, then (c) If the normals to the parabola y² = 4ax at the points t₁ & t₂ intersect again on the parabola at the point 't₃' then t₁ t₂ = 2 ; t₃ = − (t₁ + t₂) and the line joining t₁ & t₂ passes through a fixed point (−2a, 0).
General Note :
(i) Length of subtangent at any point P(x, y) on the parabola y² = 4ax equals twice the abscissa of the point P. Note that the subtangent is bisected at the vertex.
(ii) Length of subnormal is constant for all points on the parabola & is equal to the semi latus rectum.
(iii) If a family of straight lines can be represented by an equation λ2P + λQ + R = 0 where λ is a parameter and P, Q, R are linear functions of x and y then the family of lines will be tangent to the curve Q² = 4 PR.
The equation to the pair of tangents which can be drawn from any point (x₁, y₁) to the parabola y² = 4ax is given by : SS₁ = T² where :
S ≡ y² − 4ax ; S₁ = y₁² − 4ax1; T ≡ y y₁ − 2a(x + x₁).

Director Circle :
Locus of the point of intersection of the perpendicular tangents to the parabola y² = 4ax is called the Director Circle. It’s equation is x + a = 0 which is parabola’s own directrix.

Chord Of Contact :
Equation to the chord of contact of tangents drawn from a point P(x₁, y₁) is yy₁ = 2a (x + x₁).
Remember that the area of the triangle formed by the tangents from the point (x1, y1) & the chord of contact is (y₁² − 4ax₁)^3/2 ÷ 2a. Also note that the chord of contact exists only if the point P is not inside.

Polar & Pole :
(i) Equation of the Polar of the point P(x1, y1) w.r.t. the parabola y² = 4ax is,
y y₁= 2a(x + x₁)
(ii) The pole of the line lx + my + n = 0 w.r.t. the parabola y² = 4ax is Note:
(i) The polar of the focus of the parabola is the directrix.
(ii) When the point (x₁, y₁) lies without the parabola the equation to its polar is the same as the equation to the chord of contact of tangents drawn from (x₁, y₁) when (x₁, y₁) is on the parabola the polar is the same as the tangent at the point.
(iii) If the polar of a point P passes through the point Q, then the polar of Q goes through P.
(iv) Two straight lines are said to be conjugated to each other w.r.t. a parabola when the pole of one lies on the other.
(v) Polar of a given point P w.r.t. any Conic is the locus of the harmonic conjugate of P w.r.t. the two points is which any line through P cuts the conic.

Chord With A Given Middle Point :
Equation of the chord of the parabola y² = 4ax whose middle point is
(x₁, y₁) is y − y₁ (x − x₁₎. This reduced to T = S₁
where T ≡ y y₁ − 2a (x + x₁) & S₁ ≡ y₁² − 4ax₁.

Diameter :
The locus of the middle points of a system of parallel chords of a Parabola is called a Diameter. Equation to the diameter of a parabola is y = 2a/m, where m = slope of parallel chords.
Note:
(i) The tangent at the extremity of a diameter of a parabola is parallel to the system of chords it bisects.
(ii) The tangent at the ends of any chords of a parabola meet on the diameter which bisects the chord.
(iii) A line segment from a point P on the parabola and parallel to the system of parallel chords is called the ordinate to the diameter bisecting the system of parallel chords and the chords are called its double ordinate.

IMPORTANT HIGHLIGHTS :
(a) If the tangent & normal at any point ‘P’ of the parabola intersect the axis at T & G then ST = SG = SP where ‘S’ is the focus. In other words the tangent and the normal at a point P on the parabola are the bisectors of the angle between the focal radius SP & the perpendicular from P on the directrix. From this we conclude that all rays emanating from S will become parallel to the axis of the parabola after reflection.
(b) The portion of a tangent to a parabola cut off between the directrix & the curve subtends a right angle at the focus.
(c) The tangents at the extremities of a focal chord intersect at right angles on the directrix, and hence a circle on any focal chord as diameter touches the directrix. Also a circle on any focal radii of a point P (at², 2at) as diameter touches the tangent at the vertex and intercepts a chord of length a on a normal at the point P.

(d) Any tangent to a parabola & the perpendicular on it from the focus meet on the tangtent at the vertex.
(e) If the tangents at P and Q meet in T, then :

• TP and TQ subtend equal angles at the focus S.
• ST² = SP. SQ &
• The triangles SPT and STQ are similar.

(f) Tangents and Normals at the extremities of the latus rectum of a parabola y² = 4ax constitute a square, their points of intersection being (−a, 0) & (3 a, 0).
(g) Semi latus rectum of the parabola y² = 4ax, is the harmonic mean between segments of any focal chord of the parabola is ; 2a

(h) The circle circumscribing the triangle formed by any three tangents to a parabola passes through the focus.
(i) The orthocentre of any triangle formed by three tangents to a parabola y2 = 4ax lies on the directrix & has the co-ordinates − a, a (t₁ + t₂ + t₃ + t₁t₂t₃).
(j) The area of the triangle formed by three points on a parabola is twice the area of the triangle formed by the tangents at these points.

(k) If normal drawn to a parabola passes through a point P(h, k) then k = mh − 2am − am³ i.e. am³ + m(2a − h) + k = 0. Then gives m₁ + m₂ + m₃ = 0 ; m₁m₂ + m₂m₃ + m₃m₁ = ; m₁ m₂ m₃ =  where m₁, m₂, & m₃ are the slopes of the three concurrent normals. Note that the algebraic sum of the:

• slopes of the three concurrent normals is zero.
• ordinates of the three conormal points on the parabola is zero.
• Centroid of the Δ formed by three co-normal points lies on the x-axis.

(l) A circle circumscribing the triangle formed by three co−normal points passes through the vertex of the parabola and its equation is, 2(x² + y²) − 2(h + 2a)x − ky = 0
Suggested problems from S.L.Loney: Exercise-25 (Q.5, 10, 13, 14, 18, 21), Exercise-26 (Important) (Q.4, 6, 7, 16, 17, 20, 22, 26, 27, 28, 34, 38), Exercise-27 (Q.4, 7), Exercise-28 (Q.2, 7, 11, 14, 17, 23), Exercise-29 (Q.7, 8, 10, 19, 21, 24, 26, 27), Exercise-30 (2, 3, 13, 18, 20, 21, 22, 25, 26, 30)