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23 Nov Parabola

24 Nov Ellipse and Hyperbola

25 Nov 3D | Mathematical Reasoning

28 Nov Limits and Derivatives

29 Nov Statistics

30 Nov Probability
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Q. Let P(a, b) be a point on the parabola
y2 = 8x such that the tangent at P passes
through the centre of the circle
x2 + y2 - 10x - 14y + 65 = 0. Let A be the
product of all possible values of a and B be
the product of all possible values of b. Then
the value of A + B is equal to:
JEE Main 2022 27 July Shift I

A 0 B 25

C 40 D 65
Q. Let P(a, b) be a point on the parabola
y2 = 8x such that the tangent at P passes
through the centre of the circle
x2 + y2 - 10x - 14y + 65 = 0. Let A be the
product of all possible values of a and B be
the product of all possible values of b. Then
the value of A + B is equal to:
Q. Let P(a, b) be a point on the parabola
y2 = 8x such that the tangent at P passes
through the centre of the circle
x2 + y2 - 10x - 14y + 65 = 0. Let A be the
product of all possible values of a and B be
the product of all possible values of b. Then
the value of A + B is equal to:
Solution
Q. Let P(a, b) be a point on the parabola
y2 = 8x such that the tangent at P passes
through the centre of the circle
x2 + y2 - 10x - 14y + 65 = 0. Let A be the
product of all possible values of a and B be
the product of all possible values of b. Then
the value of A + B is equal to:
JEE Main 2022 27 July Shift I

A 0 B 25

C 40 D 65
Ellipse
TWO STANDARD ELLIPSE

(a) Horizontal Ellipse:

(b) Vertical Ellipse:

STANDARD EQUATION OF AN ELLIPSE

Standard equation of an ellipse referred to its

principal axes along the co-ordinate

axes is

where a > b & b2 = a2(1 - e2)

⇒ a2 - b2 = a2e2.
Where e = eccentricity (0 < e < 1).
FOCI: S ≡ (ae, 0) & S’ ≡ (-ae, 0).
 Y
GENERAL TERMINOLOGY

(a) Equation of directrices:

B(0, b)
P(x, y)
M’ M
L1 L
X’ C X
Z’ A’ S’ (ae, 0) S A Z
(-a, 0) (ae, 0) (a, 0)
L1’ L’
B’
(0, -b)
Directrix

Directrix

Y’
 Y
GENERAL TERMINOLOGY

(b) Vertices: A’ = (-a, 0) & A = (a, 0).

B(0, b)
P(x, y)
M’ M
L1 L
C X
Z’ A’ S’ (ae, 0) S A Z
(-a, 0) (ae, 0) (a, 0)
L1’ L’
B’
(0, -b)
Directrix

Directrix

Y’
 Y
GENERAL TERMINOLOGY

(c) Major axis: The line segment A’A in which

the foci S’ & S lie is of length 2a & is called the
B(0, b) major axis (a > b) of the ellipse.
P(x, y)
M’ M
L1 L
C X
Z’ A’ S’ (ae, 0) S A Z
(-a, 0) (ae, 0) (a, 0)
L1’ L’
B’
(0, -b)
Directrix

Directrix

Y’
 Y
GENERAL TERMINOLOGY

(d) Foot of directrix: Point of intersection of

major axis with directrix is called the foot of the
B(0, b) directrix (z)
P(x, y)
M’ M
L1 L
C X
Z’ A’ S’ (ae, 0) S A Z
(-a, 0) (ae, 0) (a, 0)
L1’ L’
B’
(0, -b)
Directrix

Directrix

Y’
 Y
GENERAL TERMINOLOGY

(e) Minor Axis: The y-axis intersects the ellipse

in the points B’ ≡ (0, -b) &
B(0, b) B ≡ (0, b). The line segment B’B of length 2b(b <
P(x, y) a) is called the Minor Axis of the ellipse.
M’ M
L1 L
C X
Z’ A’ S’ (ae, 0) S A Z
(-a, 0) (ae, 0) (a, 0)
L1’ L’
B’
(0, -b)
Directrix

Directrix

Y’
 Y
GENERAL TERMINOLOGY

(f) Principal Axes: The major & minor axis

together are called Principal Axes of the
B(0, b) ellipse.
P(x, y)
M’ M
L1 L
C X
Z’ A’ S’ (ae, 0) S A Z
(-a, 0) (ae, 0) (a, 0)
L1’ L’
B’
(0, -b)
Directrix

Directrix

Y’
 Y
GENERAL TERMINOLOGY

(g) Centre: The point which bisects every chord

o f the conic drawn through it is called the
B(0, b) centre of the conic. C ≡ (0, 0) the origin is the
P(x, y) centre of the ellipse
M’ M
L1 L
C X
Z’ A’ S’ (ae, 0) S A Z
(-a, 0) (ae, 0) (a, 0)
L1’ L’
B’
(0, -b)
Directrix

Directrix

Y’
 Y
GENERAL TERMINOLOGY

(h) Diameter: A chord of the conic which passes

through the centre is called a diameter of the
B(0, b) conic.
P(x, y)
M’ M
L1 L
C X
Z’ A’ S’ (ae, 0) S A Z
(-a, 0) (ae, 0) (a, 0)
L1’ L’
B’
(0, -b)
Directrix

Directrix

Y’
 Y
GENERAL TERMINOLOGY

(i) Focal Chord: A chord which passes through a

focus is called a focal chord.
B(0, b)
P(x, y)
M’ M
L1 L
C X
Z’ A’ S’ (ae, 0) S A Z
(-a, 0) (ae, 0) (a, 0)
L1’ L’
B’
(0, -b)
Directrix

Directrix

Y’
 Y
GENERAL TERMINOLOGY

(j) Double Ordinate: A chord perpendicular to

the major axis is called a double ordinate.
B(0, b)
P(x, y)
M’ M
L1 L
C X
Z’ A’ S’ (ae, 0) S A Z
(-a, 0) (ae, 0) (a, 0)
L1’ L’
B’
(0, -b)
Directrix

Directrix

Y’
 Y
GENERAL TERMINOLOGY

(k) Latus Rectum: The focal chord

perpendicular to the major axis is called the
B(0, b) latus rectum.
P(x, y)
M’ M
L1 L
C X
Z’ A’ S’ (ae, 0) S A Z
(-a, 0) (ae, 0) (a, 0)
L1’ L’
B’
(0, -b)
Directrix

Directrix

Y’
 Y
GENERAL TERMINOLOGY
(k) Latus Rectum: The focal chord
perpendicular to the major axis is called the
latus rectum.
B(0, b) (i) Length of latus rectum (LL’)
P(x, y)
M’ M
L1 L
C X
Z’ A’ S’ (ae, 0) S A Z
(-a, 0) (ae, 0) (a, 0)
(ii) Equation of latus rectum: x = 士ae.
L1’ L’ (iii) Ends of the latus rectum are
B’
(0, -b)
Directrix

Directrix

and
Y’
 Y
GENERAL TERMINOLOGY

(l) Focal radii: SP = a - ex & S’P = a + ex

⇒ SP + S’P = 2a = Major axis.
B(0, b)
P(x, y)
M’ M
L1 L
C X
Z’ A’ S’ (ae, 0) S A Z
(-a, 0) (ae, 0) (a, 0)
L1’ L’
B’
(0, -b)
Directrix

Directrix

Y’
ANOTHER FORM OF ELLIPSE
 Z y = b/e ANOTHER FORM OF ELLIPSE
Directrix B(0, b)
(a) AA’ = Minor axis = 2a

S (0, be) (b) BB’ = Major axis = 2b

A’ (-a, 0) A(a, 0)
(c) a2 = b2(1 - e2)
X’ X
C (0, 0) (d) Latus rectum LL’ = L1L1’ = ,
L1’ L1
S’ (0, -be)
Equation y = 土 be

B’(0, -b)

Z’ y = - b/e
Directrix
Y’
 Z y = b/e ANOTHER FORM OF ELLIPSE
Directrix B(0, b)
(e) Ends of the latus rectum are:

S (0, be)

A’ (-a, 0) A(a, 0)
X’ X
C (0, 0)

L1’ L1
S’ (0, -be)
(f) Equation of directrix y = 土 b/e
B’(0, -b)
(g) Eccentricity:
Z’ y = - b/e
Directrix
Y’
SHIFTED ELLIPSE

In case the centre of the ellipse is not origin

then its equation can be taken as
 POSITION OF A POINT W.R.T. AN
ELLIPSE
S=0 Outside
The point p(x1, y1) lies outside, inside or on the
ellipse according as;
Inside
PARAMETRIC REPRESENTATION

The equations x = a cos 𝛉 & y = b sin 𝛉 together

represent the ellipse

where 𝛉 is a parameter (eccentric angle).

Note that if P(𝛉) ≡ (a cos 𝛉, b sin 𝛉) is on the
ellipse then; Q(𝛉) ≡ (a cos 𝛉, a sin 𝛉) is on the
auxiliary circle.
TANGENT TO ELLIPSE

Point Form

Equation of tangent to the given ellipse at its

point (x1, y1) is

Note

For general ellipse replace x2 by (xx1), y2 by

(yy1), 2x by (x + x1), 2y by (y + y1), 2xy by (xy1 +
yx1) & c by (c).
TANGENT TO ELLIPSE

Slope Form

Equation of tangent to the given ellipse whose

slope is ‘m’, is

Point of contact are

TANGENT TO ELLIPSE

Parametric Form

Equation of tangent to the given ellipse at its

point (a cos 𝛉, b sin 𝛉), is

Note

Point of intersection of the tangents at the

points 𝝰 & 𝛃 is
NORMAL TO ELLIPSE

Point Form

Equation of the normal to the given ellipse at

(x1, y1) is:
NORMAL TO ELLIPSE

Slope Form

Equation of normal to the given ellipse whose

slope is ‘m’ is
NORMAL TO ELLIPSE

Parametric Form

Equation of the normal to the given ellipse at

the point (acosθ, bsinθ) is
axsecθ - bycosecθ = (a2 - b2).
CHORD OF CONTACT

If PA and PB be the tangents from point P(x1,

y1) to the ellipse .

The equation of the chord of contact AB is

 PAIR OF TANGENTS

A
If P(x1, y1) be any point lies outside the ellipse

, and a pair of tangents PA, PB

P X’ X can be drawn to it from P. Then the equation of

(x1, y1) C

pair of tangents of PA and PB is SS1 = T2 where

B

Y’
CHORD WITH GIVEN MIDDLE POINT

The equation of the chord of the ellipse

whose midpoint be (x1, y1) is

T = S1 where
Hyperbola
STANDARD EQUATION

Standard equation of the hyperbola is

where b2 = a2(e2 - 1) or a2e2 = a2 + b2 i.e.

 GENERAL TERMINOLOGY
y
(a) Foci :
L S ≡ (ae, 0) & S’ ≡ (-ae, 0).
B (0, b) x = (ae, b2/a)

S’ A’ C A S
x
(-ae, 0) (-a, 0) (0, 0) (a, 0) ae, 0

B’ (0, b) (ae, -b2/a)

L’
 GENERAL TERMINOLOGY
y
(b) Equations of directrices :
L
B (0, b) x = (ae, b2/a)

S’ A’ C A S
x
(-ae, 0) (-a, 0) (0, 0) (a, 0) ae, 0

B’ (0, b) (ae, -b2/a)

L’
 GENERAL TERMINOLOGY
y
(c) Vertices :
L A ≡ (a, 0) & A’ ≡ (-a, 0).
B (0, b) x = (ae, b2/a)

S’ A’ C A S
x
(-ae, 0) (-a, 0) (0, 0) (a, 0) ae, 0

B’ (0, b) (ae, -b2/a)

L’
 GENERAL TERMINOLOGY
y
(d) Latus rectum :
L (i) Equation : x = ±ae
B (0, b) x = (ae, b2/a) (ii) Length

S’ A’ C A S
x
(-ae, 0) (-a, 0) (0, 0) (a, 0) ae, 0
(iii) Ends :

B’ (0, b) (ae, -b2/a)

L’
 GENERAL TERMINOLOGY
y
(e) (i) Transverse Axis :
L The line segment A’A of
B (0, b) x = (ae, b2/a) length 2a in which the foci S’
& S both lie is called the
Transverse Axis of the
S’ A’ C A S Hyperbola.
x
(-ae, 0) (-a, 0) (0, 0) (a, 0) ae, 0

B’ (0, b) (ae, -b2/a)

L’
 GENERAL TERMINOLOGY
y
(e) (ii) Conjugate Axis :
L The line segment B’B
B (0, b) x = (ae, b2/a) between the two points B’≡
(0, -b) & B ≡ (0, b) is called as
the Conjugate Axis of the
S’ A’ C A S Hyperbola.
x
(-ae, 0) (-a, 0) (0, 0) (a, 0) ae, 0
The Transverse Axis & the
Conjugate Axis of the
B’ (0, b) (ae, -b2/a) hyperbola are together called
the Principal axes of the
L’ hyperbola.
 GENERAL TERMINOLOGY
y
(f) Focal Property :
L The difference of the focal
B (0, b) x = (ae, b2/a) distances of any point on the
hyperbola is constant and
equal to transverse axis i.e.
S’ A’ C A S ||P S| - |P S’|| = 2a.
x The distance SS’ =
(-ae, 0) (-a, 0) (0, 0) (a, 0) ae, 0
focal length.

B’ (0, b) (ae, -b2/a)

L’
 GENERAL TERMINOLOGY
y
(g) Focal distance :
L Distance of any point P(x, y)
B (0, b) x = (ae, b2/a) on Hyperbola from foci PS =
ex - a & PS’ = ex + a.

S’ A’ C A S
x
(-ae, 0) (-a, 0) (0, 0) (a, 0) ae, 0

B’ (0, b) (ae, -b2/a)

L’
 CONJUGATE HYPERBOLA

Two hyperbolas such that transverse &

conjugate axes of one hyperbola are
(0, b) respectively the conjugate & the transverse
axes of the other are called Conjugate
Hyperbola of each other.
eg.

(-a, 0) (a, 0)

are conjugate hyperbolas of each other.

(0, -b)
CONJUGATE HYPERBOLA

Note

(i) If e1 & e2 are the eccentricities of the

hyperbola & its conjugate then
e1-2 + e2-2 = 1.
(ii) The foci of a hyperbola and its conjugate are
concyclic and form the vertices of a square.
(iii) Two hyperbolas are said to be similar if
they have the same eccentricity.
SHIFTED HYPERBOLA

In case the centre of the hyperbola is not origin

then its equation can be taken as
GENERAL NOTE

Since the fundamental equation to the

hyperbola only differs from that to the ellipse in
having -b2 instead of b2 it will be found that
many propositions for the hyperbola are
derived from those for the ellipse by simply
changing the sign of b2.
POSITION OF A POINT W.R.T.
HYPERBOLA
For standard hyperbola:

Let

its centre is (0, 0) which lies outside the

hyperbola and gives negative sign i.e.
H(0, 0) = -1

The quantity

is positive, zero or negative according as the

point (x1, y1) lies within, upon or outside the
curve.
PARAMETRIC EQUATION

The equations x = a sec θ & y = b tan θ together

represents the hyperbola

where θ is a parameter.
TANGENT TO THE HYPERBOLA

Point Form

Equation of the tangent to the given hyperbola

at the point (x1, y1) is
TANGENT TO THE HYPERBOLA

Slope Form

The equation of tangents of slope m to the

given hyperbola is

Point of contact are

TANGENT TO THE HYPERBOLA

Parametric Form

Equation of the tangent to the given hyperbola

at the point
(a secθ, b tanθ) is
NORMAL TO THE HYPERBOLA

Point Form

The equation of the normal to the given

hyperbola at the point P(x1, y1) on it is
NORMAL TO THE HYPERBOLA

Slope Form

The equation of normal of slope m to the given

hyperbola is

foot of normal are

NORMAL TO THE HYPERBOLA

Parametric Form

The equation of the normal at the point P(a

secθ, b tanθ) to the given hyperbola is
 CHORD OF CONTACT

If PA and PB be the tangents from point

A Chord of contact
P(x1, y1) to the Hyperbola ,

P then the equation of the chord of contact

B
AB is

or T = 0 at (x1, y1).
PAIR OF TANGENTS

If P(x1, y1) be any point lies outside the

Hyperbola and a pair of

tangents PA, PB can be drawn to it from P.

Then the equation of pair of tangents of PA and
PB is SS1 = T2 where

i.e.
CHORD WITH GIVEN MIDDLE POINT

The equation of the chord of the hyperbola

, whose midpoint be (x1, y1)

is T = S1
where
 RECTANGULAR HYPERBOLA
y
A hyperbola is said to be rectangular hyperbola
A2 y=x if its transverse axis is equal to its conjugate
A1
axis i.e. b = a ⇒ e = .
y = -x
In rectangular hyperbola asymptotes are
perpendicular to each other.
45o
x
For x2 - y2 = a2 its asymptotes are y = 土 x.
 RECTANGULAR HYPERBOLA

Rotating the axes by an angle

about the same origin, equation of the

rectangular hyperbola x2 - y2 = a2 is

reduced to or

respectively. In

xy = c2 xy = -c2 xy = c2 or xy = -c2 asymptotes are coordinates

axes.
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Q. If distance between the foci of an ellipse is
6 and distance between its directrices is 12,
then length of its latus rectum is
JEE Main 2020 7 Jan Shift I

A 4

C 9

D
Q. If distance between the foci of an ellipse is
6 and distance between its directrices is 12,
then length of its latus rectum is
Solution
Q. If distance between the foci of an ellipse is
6 and distance between its directrices is 12,
then length of its latus rectum is
JEE Main 2020 7 Jan Shift I

A 4

C 9

D
Q. The locus of the mid point of the line
segment joining the point (4, 3) and the
points on the ellipse x2 + 2y2 = 4 is an ellipse
with eccentricity:
JEE Main 2022 26 June Shift 2

D
Q. The locus of the mid point of the line
segment joining the point (4, 3) and the
points on the ellipse x2 + 2y2 = 4 is an ellipse
with eccentricity:
Q. The locus of the mid point of the line
segment joining the point (4, 3) and the
points on the ellipse x2 + 2y2 = 4 is an ellipse
with eccentricity:
Solution
Q. The locus of the mid point of the line
segment joining the point (4, 3) and the
points on the ellipse x2 + 2y2 = 4 is an ellipse
with eccentricity:
JEE Main 2022 26 June Shift 2

D
Q. If a hyperbola passes through the point
P(10, 16) and it has vertices (±6, 0), then the
equation of the normal to it at P is:
JEE Main 2020 8 Jan Shift 2

A 2x + 5y = 100

B 2x + 5y = 10

C 2x - 5y = 100

D 5x + 2y = 100
Q. If a hyperbola passes through the point
P(10, 16) and it has vertices (±6, 0), then the
equation of the normal to it at P is:
Q. If a hyperbola passes through the point
P(10, 16) and it has vertices (±6, 0), then the
equation of the normal to it at P is:
Solution
Q. If a hyperbola passes through the point
P(10, 16) and it has vertices (±6, 0), then the
equation of the normal to it at P is:
JEE Main 2020 8 Jan Shift 2

A 2x + 5y = 100

B 2x + 5y = 10

C 2x - 5y = 100

D 5x + 2y = 100
Q. A hyperbola has its centre at the origin,
passes through the point (4, 2) and has
transverse axis of length 4 long the x-axis.
Then, the eccentricity of the hyperbola is:
JEE Main 2019 9 Jan Shift 2

C 2

D
Q. A hyperbola has its centre at the origin,
passes through the point (4, 2) and has
transverse axis of length 4 long the x-axis.
Then, the eccentricity of the hyperbola is:
Q. A hyperbola has its centre at the origin,
passes through the point (4, 2) and has
transverse axis of length 4 long the x-axis.
Then, the eccentricity of the hyperbola is:
Solution
Q. A hyperbola has its centre at the origin,
passes through the point (4, 2) and has
transverse axis of length 4 long the x-axis.
Then, the eccentricity of the hyperbola is:
JEE Main 2019 9 Jan Shift 2

C 2

D
Q. The acute angle between the pair of
tangent drawn to the ellipse 2x2 + 3y2 = 5
from the point (1, 3) is
JEE Main 2022 26 July Shift 2

D
Q. The acute angle between the pair of
tangent drawn to the ellipse 2x2 + 3y2 = 5
from the point (1, 3) is
Q. The acute angle between the pair of
tangent drawn to the ellipse 2x2 + 3y2 = 5
from the point (1, 3) is
Solution
Q. The acute angle between the pair of
tangent drawn to the ellipse 2x2 + 3y2 = 5
from the point (1, 3) is
JEE Main 2022 26 July Shift 2

D
Q. Let e1 and e2 be the eccentricities of the ellipse,
and the hyperbola,
respectively satisfying e1e2 = 1. If 𝛼
and β are the distances between the foci of the
ellipse and the foci of the hyperbola respectively,
then the ordered pair (𝛼, β) is equal to:

JEE Main 2021 25 Feb Shift I

A (8, 10)

D (8, 12)
Q. Let e1 and e2 be the eccentricities of the ellipse,
and the hyperbola,
respectively satisfying e1e2 = 1. If 𝛼
and β are the distances between the foci of the
ellipse and the foci of the hyperbola respectively,
then the ordered pair (𝛼, β) is equal to:
Q. Let e1 and e2 be the eccentricities of the ellipse,
and the hyperbola,
respectively satisfying e1e2 = 1. If 𝛼
and β are the distances between the foci of the
ellipse and the foci of the hyperbola respectively,
then the ordered pair (𝛼, β) is equal to:
Solution
Q. Let e1 and e2 be the eccentricities of the ellipse,
and the hyperbola,
respectively satisfying e1e2 = 1. If 𝛼
and β are the distances between the foci of the
ellipse and the foci of the hyperbola respectively,
then the ordered pair (𝛼, β) is equal to:

JEE Main 2021 25 Feb Shift I

A (8, 10)

D (8, 12)
Q. If m is the slope of a common tangent to
the curves and x2 + y2 = 12,
then 12m2 is equal to:

JEE Main 2022 26 June Shift 2

A 6

B 9

C 10

D 12
Q. If m is the slope of a common tangent to
the curves and x2 + y2 = 12,
then 12m2 is equal to:
Q. If m is the slope of a common tangent to
the curves and x2 + y2 = 12,
then 12m2 is equal to:
Solution
Q. If m is the slope of a common tangent to
the curves and x2 + y2 = 12,
then 12m2 is equal to:

JEE Main 2022 26 June Shift 2

A 6

B 9

C 10

D 12
Q. if e1 and e2 are eccentricities of the ellipse
and the hyperbola
respectively and if the point (e1, e2) lies on
ellipse 15x2 + 3y2 = k. Then find value of k

JEE Main 2020 9 Jan Shift I

A 14

B 15

C 16

D 17
Q. if e1 and e2 are eccentricities of the ellipse
and the hyperbola
respectively and if the point (e1, e2) lies on
ellipse 15x2 + 3y2 = k. Then find value of k
Solution
Q. if e1 and e2 are eccentricities of the ellipse
and the hyperbola
respectively and if the point (e1, e2) lies on
ellipse 15x2 + 3y2 = k. Then find value of k

JEE Main 2020 9 Jan Shift I

A 14

B 15

C 16

D 17
Q. If tangents are drawn to the ellipse x2 +
2y2 = 2 at all points on the ellipse other than
its four vertices then the mid points of the
tangents intercepted between the coordinate
axes lie on the curve:
JEE Main 2019 11 Jan Shift I

D
Q. If tangents are drawn to the ellipse x2 +
2y2 = 2 at all points on the ellipse other than
its four vertices then the mid points of the
tangents intercepted between the coordinate
axes lie on the curve:
Q. If tangents are drawn to the ellipse x2 +
2y2 = 2 at all points on the ellipse other than
its four vertices then the mid points of the
tangents intercepted between the coordinate
axes lie on the curve:
Solution
Q. If tangents are drawn to the ellipse x2 +
2y2 = 2 at all points on the ellipse other than
its four vertices then the mid points of the
tangents intercepted between the coordinate
axes lie on the curve:
JEE Main 2019 11 Jan Shift I

D
Q. If the normal at an end of a latus rectum of
an ellipse passes through an extremity of the
minor axis, then the eccentricity e of the
ellipse satisfies:
JEE Main 2020 6 Sep Shift I

A e2 + 2e - 1 = 0

B e2 + e - 1 = 0

C e4 + 2e2 - 1 = 0

D e4 + e2 - 1= 0
Q. If the normal at an end of a latus rectum of
an ellipse passes through an extremity of the
minor axis, then the eccentricity e of the
ellipse satisfies:
Solution
Q. If the normal at an end of a latus rectum of
an ellipse passes through an extremity of the
minor axis, then the eccentricity e of the
ellipse satisfies:
JEE Main 2020 6 Sep Shift I

A e2 + 2e - 1 = 0

B e2 + e - 1 = 0

C e4 + 2e2 - 1 = 0

D e4 + e2 - 1= 0
Q. For the hyperbola H : x2 - y2 = 1 and the
ellipse a > b > 0, let the
(1) Eccentricity of E be reciprocal of the
eccentricity of H, and
(2) The line be a common
tangent of E and H.
Then 4(a2 + b2) is equal to____.
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