PARABOLA
A. Conic section
A conic section, or conic is the locus of a point which moves in a plane so that its distance from a fixed point is in a constant ratio to its perpendicular distance from a fixed straight line.
(a) The fixed point is called the FOCUS.
(b) The fixed straight line is called the DIRECTRIX.
(c) The constant ratio is called the ECCENTRICITY denoted by `e'.
(d) The line passing through the focus & perpendicular to the directrix is called the AXIS.
(e) A point of intersection of a conic with its axis is called a VERTEX.
B. General equation of a conic : focal directrix property
The general equation of a conic with focus (p, q) & directrix lx + my + n = 0 is
(l^{2} + m^{2}) [(x  p)^{2} + (y  q)^{2}] = e^{2} (lx + my + n)^{2} ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0
C. 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 + 2 fgh  af^{2}  bg^{2}  ch^{2} = 0 & the general equation of a conic represents a pair of straight lines and if :
e > 1 the lines will be real & distinct intersecting at S.
e = 1 the lines will be coincident.
e < 1 the lines will be imaginary.
Case (ii) When the focus does not lie on the directrix
The conic represents :
a parabola hyperbola  an ellipse  a hyperbola  a rectangular 
e = 1 ; D 0  0 < e < 1 ; D 0  D 0 ; e > 1 ;  e > 1;D 0 
h^{2} = ab  h^{2} < ab  h^{2} > ab  h^{2} > ab ; a + b = 0 
D. Parabola
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 y^{2} = 4 ax. For this parabola :
(i) Vertex is (0, 0)
(ii) Focus is (a, 0)
(iii) Axis is y = 0
(iv) Directrix is x + a = 0
(a) Focal distance : The distance of a point on the parabola from the focus is called the focal distance of the point.
(b) Focal chord : A chord of the parabola, which passes through the focus is called a focal chord.
(c) Double ordinate : A chord of the parabola perpendicular to the axis of the symmetry is called a double ordinate.
(d) 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 y^{2} = 4ax.
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 said to be equal if they have the same latus rectum.
E. type of parabola
Four standard forms of the parabola are y^{2} = 4ax ; y^{2} =  4ax ; x^{2} = 4ay ; x^{2} = 4ay
Parabola  Vertex  Focus  Axis  Directrix  Length of Latus rectum  Ends of Latus rectum  Parametric eqution  Focal length 
y^{2} = 4ax  (0, 0)  (a, 0)  V = 0  x = a  4a  (a, ±2a)  (at^{2}, 2at)  x + a 
V^{2} = 4ax  (0, 0)  (a, 0)  y = 0  x = a  4a  (a, ±2a)  (at^{2}, 2at)  x  a 
x^{2} = +4ay  (0, 0)  (0, a)  x = 0  y = a  4a  (± 2a, a)  (2at, at^{2})  y + a 
x^{2} = 4ay  (0, 0)  (0, a)  x = 0  y = a  4a  (± 2a, a)  (2at,  at^{2})  y  a 
(y  k)^{2} = 4a(x  h)  (h, k)  (h + a, k)  V = k  k + a  h = 0  4a  (h + a, k ± 2a)  (h + at^{2}, k + 2at)  x  h + a 
(x  p)^{2} = 4b(y  q)  (p, q)  (p, b + q)  x = p  y + b  q = 0  4b  (p ± 2a, q + a)  (p + 2at, q + at^{2})  y  q + b 
F. Parametric representation
The simplest & the best form of representing the coordinates of a point on the parabola is (at^{2}, 2at). The equation x = at^{2} & y = 2at together represents the parabola y^{2} = 4ax, t being the parameter.
Ex.1 Find the vertex, axis, directrix, focus, latus rectum and the tangent at vertex for the parabola 9y^{2} – 16x – 12y – 57 = 0.
Sol. The given equation can be rewritten as which is of the form Y^{2 }= 4AX.
Hence the vertex is (61/16,2/3) The axis the y 2/3 = 0 ⇒ y =2/3
The directrix is X + A = 0
The focus is X = A and Y = 0
Length of the latus rectum = 4A = 16/9 .The tangent at the vertex is X = 0 ⇒ x = 61/16
Ex.2 The length of latus rectum of a parabola, whose focus is (2, 3) and directrix is the line x – 4y + 3 = 0 is
Sol. The length of latus rectum = 2 × perp. from focus to the directrix
Ex.3 Find the equation of the parabola whose focus is (–6, –6) and vertex (–2, 2).
Sol. Let S(–6, –6) be the focus and A(–2, 2) is vertex of the parabola. On SA take a point K(x_{1}, y_{1}) such that SA = AK. Draw KM perpendicular on SK. Then KM is the directrix of the parabola.
–6 + x_{1} = –4 and –6 + y_{1} = 4 or (x_{1}, y_{1}) = (2, 10)
Hence the equation of the directrix KM is y – 10 = m(x + 2) ....(1)
Also gradient of SK
x + 2y – 22 = 0 is the directrix
Next, let PM be a perpendicular on the directrix KM from any point P(x, y) on the parabola. From
SP = PM, the equation of the parabola is
or 5(x_{2} + y_{2} + 12x + 12y + 72) = (x + 2y – 22)^{2 }
or 4x_{2} + y_{2} – 4xy + 104x + 148y – 124 = 0
or (2x – y)^{2} + 104x + 148y – 124 = 0.
Ex.4 The extreme points of the latus rectum of a parabola are (7, 5) and (7, 3). Find the equation of the parabola
Sol. Focus of the parabola is the midpoint of the latus rectum.
⇒ S is (7, 4). Also axis of the parabola is perpendicular to the latus rectum and passes through the focus. Its equation is y – 4 =
Length of the latus rectum = (5 – 3) = 2
Hence the vertex of the parabola is at a distance 2/4 = 0.5 from the focus. We have two parabolas, one concave rightward and the other concave leftward
The vertex of the first parabola is (6.5, 4) and its equation is (y – 4)^{2} = 2(x – 6.5) and it meets the xaxis at (14.5, 0). The equation of the second parabola is (y – 4)^{2} = –2(x – 7.5). It meets the xaxis at (–0.5, 0)
G. POSITION OF A POINT RELATIVE TO A PARABOLA
The point (x_{1}, y_{1}) lies outside, on or inside the parabola y^{2} = 4ax according as the expression y^{2}  4axi is positive, zero or negative
Ex.5 Find the value of a for which the point (α –1, α ) lies inside the parabola y^{2} = 4x.
Sol. Q Point (α – 1, α) lies inside the parabola y^{2} = 4x
y_{1}^{2}  4 x_{1 }< 0 ⇒ α^{2} 4(α 1) < 0
⇒ α^{2}  4α + 4 < 0 ⇒ (α  2)^{2} < 0 ⇒ α∈ Ø
H. CHORD JOINING TWO POINTS
The equation of a chord of the parabola y^{2} = 4ax joining its two points P(t_{1}) and Q(t_{2}) is y(t_{1} + t_{2}) = 2x + 2at_{1}t_{2}
Note : (i) If PQ is focal chord then t_{1}t_{2} = –1.
(ii) Extremities of focal chord can be taken as (at^{2}, 2at) &
Ex.6 Through the vertex O of a parabola y^{2} = 4x chords OP and OQ are drawn at right angles to one another. Show that for all position of P, PQ cuts the axis of the parabola at a fixed point.
Sol. The given parabola is y^{2} = 4x ....(1)
variable line PQ passes through a fixed point which is point of intersection of L_{1} = 0 & L_{2} = 0 i.e. (4, 0)
I. LINE & A PARABOLA
(a) The line y = mx + c meets the parabola y^{2} = 4ax in two points real, coincident or imaginary according as a > = < cm ⇒ condition of tangency is, c = a/m.
Note : Line y = mx + c will be tangent to parabola x^{2} = 4ay if c = – am^{2}
(b) Length of the chord intercepted by the parabola y^{2} = 4ax on the line y = mx + c is
Note : length of the focal chord making an angle a with the xaxis is 4a cosec^{2}α
Ex.7 If the line y = 3x + λ intersect the parabola y^{2} = 4x at two distinct points then set of values of λ is
Sol. Putting value of y from the line in the parabola –
(3x + λ)^{2} = 4x ⇒ 9x^{2} + (6λ – 4)x + λ^{2} = 0
line cuts the parabola at two distinct points
D > 0 ⇒ 4(3λ – 2)^{2} – 4.9λ^{2} > 0 ⇒ 9λ^{2}  12λ + 4  9λ^{2} > 0 ⇒ λ < 1/3 Hence , λ E (∝, 1/3)
J. TANGENT TO THE PARABOLA y^{2} = 4ax
(a) Point form : Equation of tangent to the given parabola at its point (x_{1}, y_{1}) is yy_{1} = 2a (x + x_{1})
(b) Slope form : Equation of tangent to the given parabola whose slope is ‘m’, is y = mx + a/m, (m≠0) & Point of contact is
(c) Parametric form : Equation of tangent to the given parabola at its point P(t), is ty = x + at^{2}
Note : Point of intersection of the tang ents at the point t_{1} & t_{2} is [at_{1} t_{2}, a(t_{1 }+ t_{2})], (i.e. G.M. and A.M. of abscissa and ordinates of the points)
Ex.8 A tangent to the parabola y^{2} = 8x makes an angle of 45º with the straight line y = 3x + 5. Find its equation and its point of contact.
Sol. Let the slope of the tangent be m
As we know that equation of tangent of slope m to the parabola y^{2} = 4ax is y = mx + a/m. and point of contact is
for m = –2, equation of tangent is y = –2x – 1 and point of contact is (1/2,2)
for m = 1/2, equation of tangent is y = 1/2x+4 and point of contact is (8, 8)
Ex.9 Find the equation to the tangents to the parabola y^{2} = 9x which go through the point (4, 10).
Sol. Equation of tangent to parabola y^{2} = 9x is y = mx + 9/4m
Since it passes through (4, 10)
∴ 10 = 4m + 9/4m
16m^{2} – 40 m + 9 = 0
= m  1/4,9/4
∴equation of tangent’s are
Ex.10 Find the locus of the point P from which tangents are drawn to the parabola y2 = 4ax having slopes m_{1} and m_{2} such that
where θ_{1} and θ_{2} are the inclinations of the tangents from positive xaxis.
Sol. Equation of tangent to y^{2} = 4ax is y = mx + a/m
Let it passes through P(h, k).
∴ m^{2}h – mk + a = 0
locus of P(h, k) is y^{2} – 2ax = λx^{2}
K. DIRECTOR CIRCLE
Locus of the point of intersection of the perpendicular tangents to the parabola y^{2} = 4ax is called the DIRECTOR CIRCLE. It’s equation is x + a = 0 which is parabola’s own directrix.
Ex.11 The angle between the tangents drawn from a point (–a, 2a) to y^{2 }= 4ax is
Sol. The given point (–a, 2a) lies on the directrix x = –a of the parabola y^{2} = 4ax. Thus, the tangents are at right angle.
Ex.12 The circle drawn with variable chord x + ay – 5 = 0 (a being a parameter) of the parabola y^{2} = 20x as diameter will always touch the line
Sol. Clearly x + ay – 5 = 0 will always pass through the focus of y^{2} = 20x i.e. (5, 0). Thus the drawn circle will always touch the directrix of the parabola i.e.. the line x + 5 = 0.
L. NORMAL TO THE PARABOLA y^{2} = 4ax
(a) Point form : Equation of normal to the given parabola at its point (x_{1}, y_{1}) is
(b) Slope form : Equation of normal to the given parabola whose slope is ‘m’, is
y = m x – 2am – am^{3} & foot of the normal is (am^{2}, – 2am)
(c) Parametric form : Equation of normal to the given parabola at its point P(t), is y + tx = 2at + at^{3}
Note :
(i) Point of intersection of normals at t_{1} & t_{2} is,
(ii) I f the normal to the parabola y^{2} = 4 ax at the point t_{1}, meets the para bola again at the point t_{2}
(iii)If the normals to the parabola y^{2} = 4ax at the points t_{1} & t_{2} intersect again on the parabola at the point ‘t_{3}’ then t_{1}t_{2} = 2; t_{3 }= – (t_{1} + t_{2}) and the line joining t_{1} & t_{2} passes through a fixed point (–2a, 0).
(iv) If normal drawn to a parabola passes through a point P(h, k) then k = mh – 2 am – am^{3} i.e. am^{3} + m (2a – h) + k = 0.
This gives m_{1} + m_{2} + m_{3} = 0 ; m_{1}m_{2} + m_{2}m_{3} + m_{3}m_{1} =
where m_{1}, m_{2}, & m_{3} are the slopes of the three concurrent normals :
Ex.13 Prove that the normal chord to a parabola y^{2} = 4ax at the point whose ordinate is equal to abscissa subtends a right angle at the focus.
Sol. Let the normal at P (at_{1}^{2} , 2at_{1 }) meet the curve at Q (a t_{ 2}^{2} , 2 a t_{2} )
PQ is a normal chord and t_{2} = t_{1} 2/t_{1} ....(i)
By given condition 2at_{1} = at_{1}^{2}
t_{1} = 2 from equation (i), t_{2} = –3
then P(4a, 4a) and Q(9a, –6a) but focus S(a, 0)
Ex.14 If two normals drawn from any point to the parabola y^{2} = 4ax make angle a and b with the axis such that tan α . tan β = 2, then find the locus of this point,
Sol. Let the point is (h, k). The equation of any normal to the parabola y^{2} = 4ax is y = mx – 2am – am^{3} passes through (h, k) ⇒ k = mh – 2am – am^{3}
⇒ am^{3} + m(2a – h) + k = 0 ...(i)
m_{1}, m_{2}, m_{3} are roots of the equation, then m_{1.} m_{2}. m_{3 }
⇒ k^{2} = 4ah. Thus locus is y^{2} = 4ax.
Ex.15 Three normals are drawn from the point (14, 7) to the curve y^{2} – 16x – 8y = 0. Find the coordinates of the feet of the normals.
Sol. The given parabola is y^{2 }– 16x – 8y = 0 .....(i)
Let the coordinates of the feet of the normal from (14, 7) be P(α, β). Now the equation of the tangent at P(α, β) to parabola (i) is
yb – 8(x + a) – 4(y + b) = 0 or (b – 4)y = 8x + 8a + 4b ...(ii)
Its slope = 8/β4 ....(2)
Equation of the normal to parabola (i) at
It passes through (14, 7)
Also (a, b) lies on parabola (i) i.e. b^{2}  16a  8b = 0
Putting the value of a from (iii) in (iv), we get
⇒ β^{2}(β  4)  96β  8β(β  4) = 0
⇒ β(β^{2}  4β  96  8β + 32) = 0
⇒ β(β^{2}  12β  64) = 0
⇒ β(β  16)(β + 4) = 0 ⇒ β = 0, 16,  4
from (iii), a = 0 when β = 0; α = 8, when β = 16 ; α = 3 when β = –4
Hence the feet of the normals are (0, 0) (8, 16) and (3, –4)
M. LENGTH OF SUBTANGENT & SUBNORMAL
PT and PG are the tangent and normal respectively at the point P to the parabola y^{2} = 4ax. Then
TN = length of subtangent = twice the abscisse of the point P (Subtangent is always bisected by the vertex)
NG = length of subnormal which is constant for all points on the parabola & equal to its semi latusrectum (2a).
N. PAIR OF TANGENTS
The equation of the pair of tangents which can be drawn from any point P(x_{1}, y_{1}) out side the parabola to the parabola y^{2} = 4ax is given by : SS_{1} = T_{2} where :
S ≡ y^{2} – 4ax ; S_{1} ≡ y_{1}^{2} – 4ax_{1} ; T≡ yy_{1} – 2a(x + x_{1}).
O. CHORD OF CONTACT
Equation of the chord of contact of tangents drawn from a point P(x_{1}, y_{1}) is yy_{1} = 2a(x + x_{1}) Remember that the area of the triangle formed by the tangents from the point (x_{1}, y_{1}) & the chord of contact is Also note that the chord of contact exists only if the point P is not inside.
Ex.16 If the line x – y – 1 = 0 intersect the parabola y^{2} = 8x at P & Q, then find the point of intersection of tangents at P & Q
Sol. Let (h, k) be point of intersection of tangents then chord of contact is
yk = 4(x + h) ⇒ 4x – yk + 4h = 0 ............(i)
But given line is ⇒ x – y – 1 = 0 ............(ii)
Comparing (i) and (ii),
h = – 1, k = 4
Therefore, point ≡ (–1, 4)
Ex.17 Find the locus of point whose chord of contact w.r.t. to the parabola y^{2} = 4bx is the tangent of the parabola y^{2} = 4ax
Sol. Equation of tangent to y^{2} = 4ax is y = mx + a/m ....(1)
Let it is chord of contact for parabola y^{2} = 4bx w.r.t. the point P(h, k)
Equation of chord of contact is yk = 2b(x + h)
From (i) & (ii),
P. CHORD WITH A GIVEN MIDDLE POINT
Equation of the chord of the parabola y^{2} = 4ax whose middle point is (x_{1}, y_{1}) is y – y_{1} = 2a/y_{1} (xx_{1})
The reduced to T = S_{1} where T ≡ yy_{1} – 2a (x + x_{1}) & S_{1} ≡ y_{1}^{2} – 4ax_{1}.
Ex.18 Find the locus of middle of the chord of the parabola y^{2} = 4ax which pass through a given (p, q).
Sol. Let P(h, k) be the mid point of chord of the parabola y^{2} = 4ax.
so equation of chord is yk – 2a(x + h) = k^{2} – 4ah. Since it passes through (p, q)
qk – 2a(p + h) = k^{2} – 4ah
Required locus is y^{2} – 2ax – qy + 2ap = 0.
Ex.19 Find the locus of the middle point of a chord of a parabola y^{2} = 4ax which subtends a right angle at the vertex.
Sol. The equation of the chord of the parabola whose middle point is (α, β) is
points of intersection P and Q of the chord with the parabola y^{2} = 4ax is obtained by making the equation homogeneous by means of (i). Thus the equation of lines OP and OQ is
If the lines OP and OQ are at right angles, then the coefficient of x^{2} + the coefficient of y^{2} = 0. Therefore, β^{2}  2aα + 8a^{2 }= 0 ⇒ b^{2} = 2a(α  4a). Hence the locus of (a, β) is y^{2} = 2a(x  4a)
Q. AN IMPORTANT CONCEPT
If a family of straight lines can be represented by an equation λ^{2}P + λQ + R = 0 where l 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 Q^{2} = 4PR.
Ex.20 If the equation m^{2}(x + 1) + m(y – 2) + 1 = 0 represents a family of lines, where ‘m’ is parameter then find the equation of the curve to which these lines will always be tangents.
Sol. m^{2}(x + 1) + m(y – 2) + 1 = 0. The equation of the curve to which above lines will always be tangents can be obtained by equating its discriminant to zero.
(y – 2)^{2} – 4(x + 1) = 0
⇒ y^{2} – 4y + 4 – 4x – 4 = 0
⇒ y^{2} = 4(x + y)
R. DIAMETER
The locus of the middle points of a system of parallel chords of a Parabola is called DIAMETER. Equation to the diameter of a parabola is y = 2a/m, where m = slope of parallel chords.
Ex.21 The common tangent of the parabola y^{2} = 8ax and the circle x^{2} + y^{2} = 2a^{2} is
Sol. Any tangent to parabola is y = mx + 2a/m
B^{2}  4AC, gives m = +_{} 1,
Tangent y = +_{}x,+_{}a
Ex.23 If P(–3, 2) is one end of the focal chord PQ of the parabola y^{2} + 4x + 4y = 0, then the slope of the normal at Q is
Sol. The equation of the tangent at (–3, 2) to the parabola y^{2} + 4x + 4y = 0 is
2y + 2(x – 3) + 2(y + 2) = 0 or 2x + 4y – 2 = 0 = x + 2y – 1 = 0
Since the tangent at one end of the focal chord is parallel to the normal at the other end, the slope of the normal at the other end of the focal chord is =1/2
Ex.25 If r_{1}, r_{2} be the length of the perpendicular chords of the parabola y^{2} = 4ax drawn through the vertex, then show that
Sol. Since chord are perpendicular, therefore if one makes an angle θ then the other will make an angle (90º – θ) with xaxis
Let AP = r_{1} and AQ = r_{2}
If ∠PAX = θ then ∠QAX = 90º – θ
Coordinates of P and Q are (r_{1} cosq, r_{1} sinq) and (r_{2} sinθ, – r_{2} cosθ) respectively.
Since P and Q lies on y^{2} = 4ax
S. IMPORTANT HIGHLIGHTS OF PARABOLA :
(1) If the tangent & normal at any point ‘P’ of the para bola 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.
(2) The portion of a tangent to a parabola cut off between the directrix & the curve subtends a right angle a the focus.
(3) The tangents at the extremities of a focal chord intersect at right angles on the directrix, and a circle on any focal chord as diameter touches the directrix. Also a circle on any focal radii of a point P (at^{2}, 2at) as diameter touches the tangent at the vertex and intercepts a chord of length on a normal at the point P.
(4) Any tangent to a parabola & the perpendicular on it from the focus meet on the tangent at the vertex.
(5) If the tangents at P and Q meet in T, then
(i) TP and TQ subtend equal angles at the focus S
(ii) ST^{2} = SP.SQ, and
(iii)the triangles SPT and STQ are similar
Proof. Let P be the point (at^{2}, 2at) and Q be the point (at_{2}^{2} 2at_{2} ) . Coordinates of T which is the point of intersection of tangents at P and Q is {at_{1}t_{2}, a(t_{1} + t_{2})}
(i) The equation of SP is y
The perpendicular distance TU, from T on the straight line
Similarly TU has the same numerical value. The angles PST and QST are therefore equal.
(ii) We have SP = a (1 + t_{1}^{2} ) and SQ = a (1 + t_{2}^{2})
Also ST^{2 }= (at_{1}t_{2} – a)^{2 }+ a_{2} (t_{1} + t_{2})^{2}
Hence ST^{2} = SP . SQ.
(iii)Since ST/SP = SQ/ST and the angles TSP and TSQ are equal, the triangles SPT and STQ are similar, so that ∠SQT = ∠STP and∠STQ = ∠SPT.
(6) Tangents and normals at the extremities of the latus rectum of a parabola y^{2} = 4ax constitute a square, their point of intersection being (–a, 0) & (3a, 0)
(7) Semi latus rectum of the parabola y^{2} = 4ax, is the harmonic mean between segments of any focal chord of the parabola is
(8) The circle circumscribing the D formed by any 3 tangents to a parabola passes through the focus.
(9) The orthocentre of any triangle formed by three tangents to a parabola lies on the directrix.
Proof. Let the equations to the three tangents be _{t1}y = x + at_{1}^{2} .....(i)
t_{2}y = x + at_{2}^{2} ....(ii) and
t_{3}y = x + at_{3}^{2} ......(iii)
The point of intersection of (ii) and (iii) is found, by solving them, to be (at_{2}t_{3}, a(t_{2} + t_{3})) The equation to the straight line through this point perpendicular to (i) is
y – a(t_{2}+ t_{3}) = –t_{1}(x – at_{2}t_{3}) i.e. y + t_{1}x = a(t_{2} + t_{3} + t_{1}t_{2}t_{3}) ...(iv)
Similarly, the equation to the straight line through the line intersection of (iii) and (i) perpendicular to (ii) is y + t_{2}x = a(t_{3} + t_{1} + t_{1}t_{2}t_{3}) ...(v)
and the equation to the straight line through the intersection of (i) and (ii) perpendicular to (iii) is y + t_{1}x = a(t_{1} + t_{2} + t_{1}t_{2}t_{3}) (vi)
The point which is common to the straight lines (iv), (v) and (vi)
i.e. the orthocentre of the triangle, is easily seen to be the point whose coordinates are x = –a, y = a(t_{1} + t_{2} + t_{3} + t_{1}t_{2}t_{3}) and this point lies on the directrix.
(10) The area of the D formed by 3 points on a parabola is twice the area of the D formed by the tangents at these points
Proof. Let the three points on the parabola be (at_{1}^{2}, 2at_{1}) , (at_{2}^{2} 2at_{ 2} ) and (at_{3}^{2} 2at_{3} )
The area of the triangle formed by these points
The points of intersection of the tangents at these points are
(at_{2}t_{3}, a(t_{2} + t_{3})), (at_{3}t_{1}, a(t_{3} + t_{1})) and (at_{1}t_{2}, a(t_{1} + t_{2}))
The area of the triangle formed by these three points
(11) A circle circumscribing the D formed by the 3 conormal points. normals at which meet at (h, k) passes through the vertex of the parabola and its equation is 2(x^{2 }+ y^{2}) – 2(h + 2a) x – ky = 0
85 videos243 docs99 tests

Ellipse : Major and Minor Axis Relationship between Semi Major Semi Minor Axis Distance of the Focus Video  09:18 min 
Test: Circle 2 Test  30 ques 
NCERT Exemplar: Conic Sections (Exercise) Doc  33 pages 
1. What is a parabola? 
2. What is the equation of a parabola? 
3. What are some reallife examples of parabolas? 
4. What is the focus of a parabola? 
5. How is the vertex of a parabola calculated? 
Ellipse : Major and Minor Axis Relationship between Semi Major Semi Minor Axis Distance of the Focus Video  09:18 min 
Test: Circle 2 Test  30 ques 
NCERT Exemplar: Conic Sections (Exercise) Doc  33 pages 

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