Topics Completed (30 Day Board to JEE Series):
1.Indefinite Integration (Board Focus - one shot)
2.Indefinite Integration (JEE Focus - one shot)
3.Binomial Theorem (JEE Focus - one shot)
4.Definite Integration (Board Focus - one shot)
5.Definite Integration (JEE Focus - one shot)
6.Areas of Bounded Regions(Board Focus - one shot)
7.Areas of Bounded Regions(JEE Focus - one shot)
8.Differential Equations(Board + JEE Focus - one shot)
9.Application of Derivatives(Board Focus - one shot)
10.Application of Derivatives(JEE Focus - one shot)
11.Straight Lines (JEE Focus - one shot)
12.Circles (JEE Focus - one shot)
13.Parabola (JEE Focus - one shot)
Next Topics:
1. Conics (JEE Focus - one shot)
Ellipse and Hyperbola
CONIC SECTIONS
Ellipse and Hyperbola
Standard ellipses having centre at origin
B(0, b)
S’(-ae, 0)
S(ae, 0)
O
A’(-a, 0)
A(a, 0)
Y
B(0, b)
S(0, be)
X
O
A’(-a, 0)
A(a, 0)
S’(0, -be)
B’(0, -b)
Standard hyperbolas having centre at origin
Y
B(0, b)
S’(-ae, o)
Z’
A’(-a, 0)
O
Z
B’(0, -b)
A(a, 0)
S(ae, 0)
X
Y
S(0, be)
B(0, b)
Z
A’(-a, 0)
O
A(a, 0)
Z’
B’(0, -b)
S’(0, -be)
X
If a vertex of an ellipse centered at origin is (4, 0) and
the corresponding focus is (3, 0), then find its
equation.
If a vertex of an ellipse centered at origin is (4, 0) and
the corresponding focus is (3, 0), then find its
equation.
Solution
Find eccentricity of an ellipse if length of its minor axis
equal to its focal length (distance between foci).
Find eccentricity of an ellipse if length of its minor axis
equal to its focal length (distance between foci).
Solution
B
S’
S
B’
JEE Main 27th July, 2022 Shift- 1
If the length of the latus rectum of the ellipse
x2 + 4y2 + 2x + 8y - λ = 0 is 4, and l is the length of its
major axis, then λ + l is equal to
Solution
JEE Main 2020
If the distance between the foci of an ellipse is 6 and
the distance between its directrices is 12, then the
length of its latus rectum is:
A
B
C
D
JEE Main 2020
If the distance between the foci of an ellipse is 6 and
the distance between its directrices is 12, then the
length of its latus rectum is:
Solution
JEE Main 2020
If the distance between the foci of an ellipse is 6 and
the distance between its directrices is 12, then the
length of its latus rectum is:
A
B
C
D
Ellipse and Hyperbola
Q. The equation of the circle passing through the foci of the ellipse
and having centre at (0, 3) is
A
B
C
D
JEE (Main) 2013
Ellipse and Hyperbola
Q. The equation of the circle passing through the foci of the ellipse
and having centre at (0, 3) is
JEE (Main) 2013
Ellipse and Hyperbola
Q. The equation of the circle passing through the foci of the ellipse
and having centre at (0, 3) is
A
B
C
D
JEE (Main) 2013
Ellipse and Hyperbola
JEE Main 2020
If e1 and e2 are the eccentricities of the ellipse,
and the hyperbola,
respectively and (e1, e2) is a point on the ellipse,
15x2 + 3y2 = k, then k is equal to
A
14
B
15
C
16
D
17
JEE Main 2020
If e1 and e2 are the eccentricities of the ellipse,
and the hyperbola,
respectively and (e1, e2) is a point on the ellipse,
15x2 + 3y2 = k, then k is equal to
Solution
JEE Main 2020
If e1 and e2 are the eccentricities of the ellipse,
and the hyperbola,
respectively and (e1, e2) is a point on the ellipse,
15x2 + 3y2 = k, then k is equal to
A
14
B
15
C
16
D
17
JEE Main 2019
Let the length of the latus rectum of an ellipse with its
major axis along x-axis and centre at the origin, be 8.
If the distance between the foci of this ellipse is equal
to the length of its minor axis, then which one of the
following points lies on it?
A
B
C
D
JEE Main 2019
Let the length of the latus rectum of an ellipse with its
major axis along x-axis and centre at the origin, be 8.
If the distance between the foci of this ellipse is equal
to the length of its minor axis, then which one of the
following points lies on it?
Solution
JEE Main 2019
Let the length of the latus rectum of an ellipse with its
major axis along x-axis and centre at the origin, be 8.
If the distance between the foci of this ellipse is equal
to the length of its minor axis, then which one of the
following points lies on it?
A
B
C
D
Standard hyperbolas having centre at origin
Y
B(0, b)
S’(-ae, o)
Z’
A’(-a, 0)
O
Z
B’(0, -b)
A(a, 0)
S(ae, 0)
X
Y
S(0, be)
B(0, b)
Z
A’(-a, 0)
O
A(a, 0)
Z’
B’(0, -b)
S’(0, -be)
X
Find the equation of the hyperbola whose foci are
(8, 3), (0, 3) and eccentricity
Find the equation of the hyperbola whose foci are
(8, 3), (0, 3) and eccentricity
Solution
Solution
JEE Main 2019
If 5x + 9 = 0 is the directrix of the hyperbola
16x2 - 9y2 = 144, then its corresponding focus is:
A
(5, 0)
B
C
D
(-5, 0)
JEE Main 2019
If 5x + 9 = 0 is the directrix of the hyperbola
16x2 - 9y2 = 144, then its corresponding focus is:
Solution
JEE Main 2019
If 5x + 9 = 0 is the directrix of the hyperbola
16x2 - 9y2 = 144, then its corresponding focus is:
A
(5, 0)
B
C
D
(-5, 0)
Conjugate Hyperbolas
Rectangular Hyperbola
Conjugate Hyperbolas
Two hyperbolas, such that the transverse and conjugate axes
of one, are the conjugate and transverse axes of the other,
respectively, are called conjugate hyperbolas.
Result
If e1 and e2 are the eccentricities of two conjugate hyperbolas,
then
JEE Main 2014
The eccentricity of the conjugate hyperbola of the
hyperbola x2 - 3y2 = 1 is
A
2
B
C
4
D
JEE Main 2014
The eccentricity of the conjugate hyperbola of the
hyperbola x2 - 3y2 = 1 is
Solution
JEE Main 2014
The eccentricity of the conjugate hyperbola of the
hyperbola x2 - 3y2 = 1 is
A
2
B
C
4
D
Rectangular Hyperbola
If a = b, that is lengths of transverse and conjugate axes are
equal, then the hyperbola is called rectangular or equilateral.
Eg. The hyperbola x2 − y2 = a2 is a rectangular hyperbola.
Remark
1. Eccentricity of an equilateral hyperbola is always
.
Parametric forms
Position of a Point with respect to a Conic
Result
For Ellipse:
(1) S1 > 0 ⇒ point P lies outside the ellipse
(2) S1 = 0 ⇒ point P lies on the ellipse
(3) S1 < 0 ⇒ point P lies inside the ellipse
Position of a Point with respect to a Conic
Result
For Hyperbola:
(1) S1 > 0 ⇒ point lies inside hyperbola
(2) S1 = 0 ⇒ point lies on hyperbola
(3) S1 < 0 ⇒ point lies outside hyperbola
Position of a Line with respect to a Conic
Position of a Line with respect to a Conic
General Method
Solve line with conic to get a quadratic equation.
D > 0 ⇒ line cuts the conics
D = 0 ⇒ line is tangent to conics
D < 0 ⇒ line does not meet conics
Equations of Tangents of Ellipse and Hyperbola
Equations of Tangents of an Ellipse and Hyperbola
Slope form
For Ellipse
For Hyperbola
: Tangent is
: Tangent is
For Hyperbola :
Tangent of slope m is given by
Equations of Tangents of an Ellipse and Hyperbola
Slope form
For Ellipse
For Hyperbola
: Tangent is
: Tangent is
For Hyperbola :
Hence, tangent of slope m is given by
Equations of Tangents of an Circle, Ellipse and Hyperbola
JEE Main 2020
If
is a tangent to the ellipse
for some a ∈ R, then the distance between the foci of
the ellipse is:
A
B
C
4
D
JEE Main 2020
If
is a tangent to the ellipse
for some a ∈ R, then the distance between the foci of
the ellipse is:
Solution
JEE Main 2020
If
is a tangent to the ellipse
for some a ∈ R, then the distance between the foci of
the ellipse is:
A
B
C
4
D
JEE Main 2019
If the line x – 2y = 12 is tangent to the ellipse
at the point ,
the latus rectum of the ellipse is:
then the length of
Solution
JEE Main 2019
If the line x – 2y = 12 is tangent to the ellipse
at the point ,
the latus rectum of the ellipse is:
Ans: 9
then the length of
JEE Main 2015
The area (in sq units) of the quadrilateral formed by
the tangents at the end points of the latus recta of the
ellipse
A
is _____.
B
18
C
D
27
JEE Main 2015
The area (in sq units) of the quadrilateral formed by
the tangents at the end points of the latus recta of the
ellipse
is _____.
Solution
By symmetry, we can see
that the axes of the ellipse
are perpendicular bisectors
of each other.
Also, the tangents at the
extremities of the latus
recta intersect the
C
corresponding axes at
points A, B, C and D,
as shown.
So, quadrilateral ABCD is a
rhombus.
Y
B
L
O
A
D
X
Solution
JEE Main 2015
The area (in sq units) of the quadrilateral formed by
the tangents at the end points of the latus recta of the
ellipse
A
is _____.
B
18
C
D
27
JEE Main 2019
Tangent drawn to the ellipse
at point ‘P’
meets the coordinate axes at points A and B
respectively. Locus of mid-point of segment AB is
A
B
C
D
JEE Main 2019
Tangent drawn to the ellipse
at point ‘P’
meets the coordinate axes at points A and B
respectively. Locus of mid-point of segment AB is
Solution
JEE Main 2019
Tangent drawn to the ellipse
at point ‘P’
meets the coordinate axes at points A and B
respectively. Locus of mid-point of segment AB is
A
B
C
D
JEE Main 2019
If the tangents on the ellipse 4x2 + y2 = 8 at the points
(1, 2) and (a, b) are perpendicular to each other, then
a2 is equal to:
A
B
C
D
JEE Main 2019
If the tangents on the ellipse 4x2 + y2 = 8 at the points
(1, 2) and (a, b) are perpendicular to each other, then
a2 is equal to:
Solution
JEE Main 2019
If the tangents on the ellipse 4x2 + y2 = 8 at the points
(1, 2) and (a, b) are perpendicular to each other, then
a2 is equal to:
A
B
C
D
JEE Main 2020
If the line y = mx + c is a common tangent to the
hyperbola
and the circle x2 + y2 =
36, then which one of the following is true?
A
5m = 4
B
8m + 5 = 0
C
c2 = 369
D
4c2 = 369
JEE Main 2020
If the line y = mx + c is a common tangent to the
hyperbola
and the circle x2 + y2 = 36,
then which one of the following is true?
Solution
JEE Main 2020
If the line y = mx + c is a common tangent to the
hyperbola
and the circle x2 + y2 = 36,
then which one of the following is true?
A
5m = 4
B
8m + 5 = 0
C
c2 = 369
D
4c2 = 369
Equations of Normals of a Conic
Equations of Normals of an Ellipse
Equation of normal at P(x1, y1) on
P(x1, y1)
Equations of Normals of a Hyperbola
Equation of normal at P(x1, y1) on
JEE Main 2020
If a hyperbola passes through the point P(10, 16) and
it has vertices at (±6, 0), then the equation of the
normal to it at P is
A
x + 2y = 42
B
2x + 5y = 100
C
x + 3y = 58
D
3x + 4y = 94
Solution
JEE Main 2020
If a hyperbola passes through the point P(10, 16) and
it has vertices at (±6, 0), then the equation of the
normal to it at P is
A
x + 2y = 42
B
2x + 5y = 100
C
x + 3y = 58
D
3x + 4y = 94
8th Jan 2020-(Shift 1)
All the very best :-)
DREAM ON!