What is the length of the latus rectum of an ellipse?

The latus rectum is the focal chord perpendicular to the major axis; its length is 2b^2/a = 2a(1 - e^2) for the standard ellipse with a > b.

What is the eccentric angle of a point on an ellipse?

The eccentric angle theta of a point P on the ellipse is the angle at the centre corresponding to the point Q on the auxiliary circle directly above or below P. So P = (a cos theta, b sin theta).

How do I find foci and directrices from the standard equation?

Compute e = sqrt(1 - b^2/a^2) if a > b. Foci are (±ae, 0), directrices x = ±a/e. If b > a, foci are (0, ±be), directrices y = ±b/e.

Introduction to Ellipse

Q: What is an ellipse in the simplest terms?

An ellipse is a smooth closed curve that looks like a stretched circle. Formally, it is the locus of points for which the ratio of distance from a fixed focus to a fixed directrix is a constant less than 1, or equivalently, the sum of distances from two fixed foci is constant.

Q: How do I identify the major axis of an ellipse from its equation?

In x^2/p + y^2/q = 1, whichever denominator is larger tells you the major axis. If p > q, major axis is along the x-axis; if q > p, major axis is along the y-axis.

Q: What is the equation of the chord of an ellipse joining two eccentric angles?

The chord joining P(alpha) and P(beta) is (x/a) cos((alpha+beta)/2) + (y/b) sin((alpha+beta)/2) = cos((alpha-beta)/2).

Q: When does a general second-degree equation represent an ellipse?

The equation ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 represents an ellipse when h^2 < ab, Delta != 0, and 0 < e < 1.

Q: How is the auxiliary circle used in problem solving?

The auxiliary circle x^2 + y^2 = a^2 gives the clean parametrisation (a cos theta, b sin theta). Many locus and chord problems reduce to circle problems by scaling y by a/b.

Q: Is the ellipse important for JEE?

Yes. JEE Main covers standard parameters, focal chords, tangents and normals. JEE Advanced adds parametric chords, conjugate diameters, director circle and reflection properties.

MathsEllipseFor JEE aspirants

An ellipse is the locus of a point that moves so that the ratio of its distance from a fixed point (focus) to its distance from a fixed line (directrix) is a constant $e$ with $0 < e < 1$ (the eccentricity). The standard equation, referred to its principal axes as coordinate axes, is $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ , where $b^{2} = a^{2} (1 - e^{2})$ . Equivalently, an ellipse is the set of points $P$ for which $S P + S^{'} P = 2 a$ , where $S$ and $S^{'}$ are the two foci. This chapter covers the standard equation, all key parameters (foci, directrices, latus rectum, vertices), the auxiliary circle and eccentric angle, parametric representation, focal distance, and the position of a point relative to the ellipse - a compact JEE Main and Advanced foundation.

Key Formulas - Quick Reference

Standard ellipse ( $a > b$ ): $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ , with $b^{2} = a^{2} (1 - e^{2})$ .
Eccentricity: $e = 1 - \frac{b ^{2}}{a ^{2}}$ , and $0 < e < 1$ .
Foci: $S (a e, 0)$ , $S^{'} (- a e, 0)$ . Directrices: $x = \pm \frac{a}{e}$ .
Vertices: $A (a, 0)$ , $A^{'} (- a, 0)$ . Minor-axis ends: $(0, \pm b)$ .
Length of major axis $= 2 a$ ; minor axis $= 2 b$ .
Length of latus rectum: $\frac{2 b ^{2}}{a} = 2 a (1 - e^{2})$ .
Focal distances of $P (a cos θ, b sin θ)$ : $S P = a - a e cos θ$ , $S^{'} P = a + a e cos θ$ .
$S P + S^{'} P = 2 a$ (sum-of-distances definition).
Parametric form: $x = a cos θ$ , $y = b sin θ$ , $- π < θ \leq π$ .
Chord joining $α, β$ : $\frac{x}{a} cos \frac{α + β}{2} + \frac{y}{b} sin \frac{α + β}{2} = cos \frac{α - β}{2}$ .
Auxiliary circle: $x^{2} + y^{2} = a^{2}$ ; ratio $\frac{P N}{QN} = \frac{b}{a}$ .
Position: $S_{1} = \frac{x _{1}^{2}}{a ^{2}} + \frac{y _{1}^{2}}{b ^{2}} - 1$ ; $P$ is inside if $S_{1} < 0$ , on if $= 0$ , outside if $> 0$ .
General conic $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ is an ellipse iff $h^{2} < ab$ , $Δ \neq = 0$ , and $0 < e < 1$ .
Area of the ellipse region: $π ab$ .

1. Definition of an Ellipse

An ellipse is the locus of a point $P$ that moves in a plane so that the ratio of its distance from a fixed point to its perpendicular distance from a fixed line is a constant less than $1$ .

Terminology.

The fixed point is the focus.
The fixed line is the directrix (assumed not to pass through the focus, and all points and lines lie in the same plane).
The constant ratio $e$ is the eccentricity, with $0 < e < 1$ for an ellipse.

If $P$ is a moving point, $S$ the focus, and $M$ the foot of perpendicular from $P$ on the directrix, then

\frac{S P}{P M} = e, 0 < e < 1.

2. General Equation of a Conic as an Ellipse

If the focus is $(p, q)$ and the directrix is $ℓ x + m y + n = 0$ , the focus-directrix condition $S P^{2} = e^{2} \cdot P M^{2}$ gives

(ℓ^{2} + m^{2}) [(x - p)^{2} + (y - q)^{2}] = e^{2} (ℓ x + m y + n)^{2} .

Expanding, this reduces to the second-degree equation

a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0.

Ellipse condition. The general second-degree equation above represents an ellipse when

$0 < e < 1$ ,
$Δ = ab c + 2 f g h - a f^{2} - b g^{2} - c h^{2} \neq = 0$ ,
$h^{2} < ab$ .

Solved Example 1

Find the equation of the ellipse whose focus is

(- 1, 1)

, directrix is

x - y + 3 = 0

, and eccentricity

\frac{1}{2}

Solution:

Let $P (h, k)$ be a moving point. Then

\frac{S P}{P M} = \frac{1}{2} ⟹ S P^{2} = \frac{1}{4} P M^{2} .

(h + 1)^{2} + (k - 1)^{2} = \frac{1}{4} (\frac{h - k + 3}{2})^{2} .

Multiplying by $8$ and simplifying:

8 {h^{2} + k^{2} + 2 h - 2 k + 2} = (h^{2} + k^{2} - 2 hk + 6 h - 6 k + 9) .

Locus (replacing $(h, k)$ by $(x, y)$ ):

7 x^{2} + 7 y^{2} + 2 x y + 10 x - 10 y + 7 = 0.

3. Standard Equation of an Ellipse (Horizontal, $a > b$ )

Choosing the major axis along the $x$ -axis and the centre at the origin, the standard equation is

\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, a > b, b^{2} = a^{2} (1 - e^{2}) .

Figure 1: Standard horizontal ellipse

\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1

with

a > b

: centre

C

, vertices

A, A^{'}

, foci

S, S^{'}

, minor-axis endpoints

B, B^{'}

, directrices

x = \pm a / e

, and one latus rectum

L L^{'}

3.1 All the standard parameters

Quantity	Expression
Eccentricity	$e = 1 - \frac{b ^{2}}{a ^{2}}$ , $0 < e < 1$
Foci	$S (a e, 0)$ and $S^{'} (- a e, 0)$
Directrices	$x = \frac{a}{e}$ and $x = - \frac{a}{e}$
Vertices	$A (a, 0)$ and $A^{'} (- a, 0)$
Major axis	Segment $A^{'} A$ along the $x$ -axis; length $2 a$
Minor axis	Segment $B^{'} B$ along the $y$ -axis; length $2 b$ ; $B (0, b)$ , $B^{'} (0, - b)$
Principal axis	Major and minor axes together
Centre	$C (0, 0)$ ; bisects every chord through it
Focal chord	Any chord passing through a focus
Double ordinate	Chord perpendicular to the major axis
Latus rectum $L L^{'}$	Focal chord perpendicular to the major axis; length $\frac{2 b ^{2}}{a} = 2 a (1 - e^{2})$
Foot of directrix $Z$	Intersection of major axis with a directrix

The latus rectum can also be written as

2 e \times (distance from focus to the corresponding directrix)

3.2 Vertical case ( $b > a$ ): major axis on the $y$ -axis

If $b > a$ in $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ , then the $y$ -axis becomes the major axis and the $x$ -axis becomes the minor axis. All other elements swap accordingly.

Figure 2: Vertical ellipse (

b > a

): major axis along the

y

-axis, foci at

(0, \pm b e)

, directrices

y = \pm b / e

, latus rectum

L L^{'} = \frac{2 a ^{2}}{b}

, with

a^{2} = b^{2} (1 - e^{2})

Element	Formula ( $b > a$ )
Equation	$\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ , $b > a$
Relation	$a^{2} = b^{2} (1 - e^{2})$
Eccentricity	$e = 1 - \frac{a ^{2}}{b ^{2}}$
Foci	$(0, \pm b e)$
Directrices	$y = \pm \frac{b}{e}$
Vertices	$(0, \pm b)$
Length of latus rectum	$\frac{2 a ^{2}}{b}$ ; endpoints $y = \pm b e$ , $x = \pm \frac{a ^{2}}{b}$
Centre	$(0, 0)$

Convention. If the equation

\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1

is given without further specification, assume

a > b

(major axis along the

x

-axis).

4. Shifted Ellipse: Centre at $(h, k)$

An ellipse with centre $(h, k)$ and axes parallel to the coordinate axes has the equation

\frac{( x - h ) ^{2}}{a ^{2}} + \frac{( y - k ) ^{2}}{b ^{2}} = 1.

All properties transfer by the substitution $X = x - h$ , $Y = y - k$ . So the centre becomes $(h, k)$ , vertices are $(h \pm a, k)$ , foci are $(h \pm a e, k)$ , directrices are $x = h \pm \frac{a}{e}$ , and so on.

Solved Example 2

Find the equation of an ellipse whose centre is the origin, axes are the coordinate axes, and which passes through

(2, 2)

and

(3, 1)

Solution:

Let the ellipse be $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ . Substituting the two points:

\frac{4}{a ^{2}} + \frac{4}{b ^{2}} = 1 ...(i), \frac{9}{a ^{2}} + \frac{1}{b ^{2}} = 1 ...(ii) .

Computing $(i) - 4 \cdot (ii)$ :

\frac{4 - 36}{a ^{2}} = 1 - 4 ⟹ a^{2} = \frac{32}{3} .

From (i), $\frac{1}{b ^{2}} = \frac{1}{4} - \frac{3}{32} = \frac{5}{32}$ , so $b^{2} = \frac{32}{5}$ .

Ellipse: $3 x^{2} + 5 y^{2} = 32$ .

Solved Example 3

Find the equation of the ellipse whose foci are

(4, 0)

and

(- 4, 0)

and eccentricity is

\frac{1}{3}

Solution:

Foci lie on the $x$ -axis, so the $x$ -axis is the major axis and the centre is the mid-point of the foci, namely the origin. Let the ellipse be $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ .

From $a e = 4$ and $e = \frac{1}{3}$ , we get $a = 12$ , so $a^{2} = 144$ . Then

b^{2} = a^{2} (1 - e^{2}) = 144 (1 - \frac{1}{9}) = 144 \cdot \frac{8}{9} = 128.

Ellipse: $\frac{x ^{2}}{144} + \frac{y ^{2}}{128} = 1$ .

Solved Example 4

If the minor axis of an ellipse subtends a right angle at its focus, find the eccentricity.

Solution:

Let the ellipse be $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ , $a > b$ . Let $B, B^{'}$ be the ends of the minor axis and $S$ a focus. Since $∠ B S B^{'} = \frac{π}{2}$ and $O B = O B^{'}$ , we get $∠ B S O = \frac{π}{4}$ , so $O S = O B$ , i.e., $a e = b$ .

Squaring: $a^{2} e^{2} = b^{2} = a^{2} (1 - e^{2})$ , which gives $2 e^{2} = 1$ , so $e = \frac{1}{2}$ .

Solved Example 5

Find the equation of the axes, directrices, foci, centre, vertices, length of latus rectum, and eccentricity of the ellipse

\frac{( x - 3 ) ^{2}}{25} + \frac{( y - 2 ) ^{2}}{16} = 1

Solution:

Set $X = x - 3$ , $Y = y - 2$ , giving $\frac{X ^{2}}{5 ^{2}} + \frac{Y ^{2}}{4 ^{2}} = 1$ , an ellipse with $a = 5$ , $b = 4$ in $(X, Y)$ coordinates.

Major axis ( $Y = 0$ ): $y = 2$ . Minor axis ( $X = 0$ ): $x = 3$ .
Centre: $(3, 2)$ . Semi-major $a = 5$ , semi-minor $b = 4$ .
Eccentricity: $e = \frac{a ^{2} - b ^{2}}{a ^{2}} = \frac{25 - 16}{25} = \frac{3}{5}$ .
Length of latus rectum: $\frac{2 b ^{2}}{a} = \frac{32}{5}$ .
Foci: $X = \pm a e = \pm 3$ , $Y = 0$ $\Rightarrow$ $S (6, 2)$ and $S^{'} (0, 2)$ .
Vertices (major axis): $(8, 2)$ and $(- 2, 2)$ . Minor axis ends: $(3, 6)$ and $(3, - 2)$ .
Directrices: $X = \pm \frac{a}{e} = \pm \frac{25}{3}$ $\Rightarrow$ $x = \frac{34}{3}$ and $x = - \frac{16}{3}$ .

5. Auxiliary Circle and the Eccentric Angle

The circle described on the major axis of an ellipse as diameter is called the auxiliary circle. For the standard ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ , the auxiliary circle is $x^{2} + y^{2} = a^{2}$ .

Let $Q$ be a point on the auxiliary circle. Draw a perpendicular from $Q$ to the $x$ -axis; it meets the ellipse at $P$ . Then $P$ and $Q$ are called corresponding points, and the angle $θ$ that $C Q$ makes with the positive $x$ -axis (i.e., $∠ QC N$ , $- π < θ \leq π$ ) is called the eccentric angle of $P$ .

Coordinates: $Q \equiv (a cos θ, a sin θ)$ and $P \equiv (a cos θ, b sin θ)$ .

Figure 3: Auxiliary circle

x^{2} + y^{2} = a^{2}

of the ellipse. The eccentric angle

θ

of point

P

on the ellipse is measured at the centre using the corresponding point

Q

on the auxiliary circle.

Key ratio.

\frac{P N}{QN} = \frac{b}{a} = \frac{semi-minor axis}{semi-major axis}

Conversely, if perpendiculars are dropped from each point of a circle onto a fixed diameter, the locus of the points dividing these perpendiculars in a fixed ratio is an ellipse for which the given circle is the auxiliary circle.

6. Parametric Representation

The equations

x = a cos θ, y = b sin θ

together represent the standard ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ , where $θ$ is the eccentric angle parameter. If $P (θ) \equiv (a cos θ, b sin θ)$ is on the ellipse, then $Q (θ) \equiv (a cos θ, a sin θ)$ is on the auxiliary circle.

6.1 Equation of the chord joining $α$ and $β$

The chord of the ellipse joining points with eccentric angles $α$ and $β$ is

\frac{x}{a} cos \frac{α + β}{2} + \frac{y}{b} sin \frac{α + β}{2} = cos \frac{α - β}{2} .

Solved Example 6

From a point

Q

on the circle

x^{2} + y^{2} = a^{2}

, a perpendicular

QM

is drawn to the

x

-axis. Find the locus of the point

P

dividing

QM

in the ratio

2 : 1

Solution:

Let $Q \equiv (a cos θ, a sin θ)$ and $M \equiv (a cos θ, 0)$ . If $P \equiv (h, k)$ divides $QM$ in $2 : 1$ (with $P$ nearer $M$ ), then $h = a cos θ$ and $k = \frac{a sin θ}{3}$ .

Eliminating $θ$ : $(\frac{h}{a})^{2} + (\frac{3 k}{a})^{2} = 1$ .

Locus: $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{( a /3 ) ^{2}} = 1$ , an ellipse of which the given circle is the auxiliary circle.

Solved Example 7

Write the equation of the chord of the ellipse

\frac{x ^{2}}{25} + \frac{y ^{2}}{16} = 1

joining the points

P (\frac{π}{4})

and

Q (\frac{5 π}{4})

Solution:

Here $\frac{α + β}{2} = \frac{3 π}{4}$ and $\frac{α - β}{2} = - \frac{π}{2}$ , so $cos \frac{α - β}{2} = 0$ .

Chord equation: $\frac{x}{5} cos \frac{3 π}{4} + \frac{y}{4} sin \frac{3 π}{4} = 0$ , i.e., $- \frac{x}{5} + \frac{y}{4} = 0$ , or $4 x = 5 y$ .

7. Focal Distance and the "Sum of Distances" Property

For a point $P (θ) \equiv (a cos θ, b sin θ)$ on the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ (with $a > b$ ), the focal distances to the two foci $S (a e, 0)$ and $S^{'} (- a e, 0)$ are

S P = a - a e cos θ, S^{'} P = a + a e cos θ .

Adding, we get the celebrated sum-of-distances property of an ellipse:

S P + S^{'} P = 2 a = A A^{'} .

Figure 4: Focal distance property: for every point

P

on the ellipse,

S P + S^{'} P = 2 a

. This gives the alternative "sum of distances" definition of an ellipse.

Alternative definition. An ellipse can equivalently be defined as the locus of a point in a plane whose distances from two fixed points (the foci) have a constant sum.

Solved Example 8

Find the focal distances of a point

P (θ)

on the ellipse

\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1

(

a > b

Solution:

Let $M$ and $M^{'}$ be the feet of perpendiculars from $P$ on the directrices $x = \frac{a}{e}$ and $x = - \frac{a}{e}$ respectively. Then by the focus-directrix property,

S P = e \cdot P M = e (\frac{a}{e} - a cos θ) = a - a e cos θ,

S^{'} P = e \cdot P M^{'} = e (a cos θ + \frac{a}{e}) = a + a e cos θ .

Hence the focal distances are $a \pm a e cos θ$ , and $S P + S^{'} P = 2 a$ .

Solved Example 9

Find the distance from the centre of a point

P

on the ellipse

\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1

whose radius vector

C P

makes an angle

α

with the

x

-axis.

Solution:

Let $P \equiv (a cos θ, b sin θ)$ . Slope of $O P$ is $\frac{b sin θ}{a cos θ} = tan α$ , so $tan θ = \frac{a}{b} tan α$ .

Now

O P^{2} = a^{2} cos^{2} θ + b^{2} sin^{2} θ = \frac{a ^{2} + b ^{2} tan ^{2} θ}{sec ^{2} θ} = \frac{a ^{2} + b ^{2} tan ^{2} θ}{1 + tan ^{2} θ} .

Substituting $tan^{2} θ = \frac{a ^{2}}{b ^{2}} tan^{2} α$ and simplifying:

O P = \frac{ab}{a ^{2} sin ^{2} α + b ^{2} cos ^{2} α} .

8. Position of a Point with Respect to an Ellipse

For a point $P (x_{1}, y_{1})$ , define

S_{1} \equiv \frac{x _{1}^{2}}{a ^{2}} + \frac{y _{1}^{2}}{b ^{2}} - 1.

Then $P$ lies

inside the ellipse if $S_{1} < 0$ ,
on the ellipse if $S_{1} = 0$ ,
outside the ellipse if $S_{1} > 0$ .

Solved Example 10

Check whether the point

P (3, 2)

lies inside or outside the ellipse

\frac{x ^{2}}{25} + \frac{y ^{2}}{16} = 1

Solution:

S_{1} = \frac{9}{25} + \frac{4}{16} - 1 = \frac{9}{25} + \frac{1}{4} - 1 = \frac{36 + 25 - 100}{100} = - \frac{39}{100} < 0.

Since $S_{1} < 0$ , the point $P (3, 2)$ lies inside the ellipse.

Solved Example 11

Find the set of values of

α

for which the point

P (α, - α)

lies inside the ellipse

\frac{x ^{2}}{16} + \frac{y ^{2}}{9} = 1

Solution:

Require $S_{1} < 0$ :

\frac{α ^{2}}{16} + \frac{α ^{2}}{9} - 1 < 0 ⟹ \frac{25 α ^{2}}{144} < 1 ⟹ α^{2} < \frac{144}{25} .

Hence $α \in (- \frac{12}{5}, \frac{12}{5})$ .

9. Area Enclosed by the Ellipse

The area of the region bounded by the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ is

Area = π ab .

This follows from integration: $Area = 4 \int_{0}^{a} \frac{b}{a} a^{2} - x^{2} d x = π ab$ . When $a = b$ , the ellipse becomes a circle and the formula gives the familiar $π a^{2}$ .

Common Mistakes to Avoid

Watch out

Do not assume $a > b$ automatically. If the denominators of $x^{2}$ and $y^{2}$ are given as specific numbers, decide which is larger before writing foci or eccentricity. If the larger denominator is under $y^{2}$ , the major axis is on the $y$ -axis and formulas swap.
The relation is $b^{2} = a^{2} (1 - e^{2})$ only when $a > b$ . If $b > a$ , it becomes $a^{2} = b^{2} (1 - e^{2})$ .
Eccentricity of an ellipse must satisfy $0 < e < 1$ . If your calculation gives $e \geq 1$ , recheck: you may have swapped $a$ and $b$ , or the curve is not an ellipse.
The eccentric angle $θ$ of a point $P$ on the ellipse is not the polar angle of $P$ . It is the angle at the centre for the corresponding point $Q$ on the auxiliary circle, so $P = (a cos θ, b sin θ)$ , not $(r cos θ, r sin θ)$ .
The focal distance formula $S P = a - a e cos θ$ assumes the standard horizontal ellipse and the focus $S (a e, 0)$ ; if the ellipse is vertical, use the analogous formulas with $b$ in place of $a$ and $sin θ$ in place of $cos θ$ .
For a shifted ellipse, always translate first ( $X = x - h$ , $Y = y - k$ ); do not try to read foci or directrices directly off the shifted equation.
When checking the general conic condition, remember $h^{2} < ab$ is required along with $Δ \neq = 0$ . The condition $h^{2} < ab$ alone is insufficient.

Frequently Asked Questions

Q1. What is an ellipse in the simplest terms?

An ellipse is a smooth closed curve that looks like a stretched circle. Formally, it is the locus of points $P$ for which either (a) the ratio of distance from a fixed focus to a fixed directrix is a constant $e$ with $0 < e < 1$ , or equivalently, (b) the sum of distances from two fixed foci is a constant $2 a$ . Both definitions describe the same curve.

Q2. How do I identify the major axis of an ellipse from its equation?

In $\frac{x ^{2}}{p} + \frac{y ^{2}}{q} = 1$ , the larger of $p$ and $q$ tells you the major axis. If $p > q$ , the major axis is along the $x$ -axis with $a^{2} = p$ , $b^{2} = q$ . If $q > p$ , the major axis is along the $y$ -axis with $b^{2} = q$ , $a^{2} = p$ , and foci lie on the $y$ -axis.

Q3. What is the length of the latus rectum and why is it useful?

The latus rectum is the focal chord perpendicular to the major axis; its length is $\frac{2 b ^{2}}{a} = 2 a (1 - e^{2})$ . It is useful because it depends only on $a$ and $b$ (or $a$ and $e$ ), so many problems that mention it convert directly into an equation between $a$ and $b$ .

Q4. What is the eccentric angle, and how is it different from the polar angle?

The eccentric angle $θ$ of a point $P$ on the ellipse is the angle at the centre for the corresponding point $Q$ on the auxiliary circle, where $Q$ is directly above/below $P$ on the same vertical line. So $P = (a cos θ, b sin θ)$ . The polar angle of $P$ (the angle $C P$ makes with the $x$ -axis) is generally different from $θ$ unless $a = b$ .

Q5. How do I quickly find foci and directrices from the standard equation?

Identify $a^{2}$ and $b^{2}$ , then compute $e = 1 - b^{2} / a^{2}$ (assuming $a > b$ ). Foci are $(\pm a e, 0)$ and directrices are $x = \pm a / e$ . For the vertical case ( $b > a$ ), foci are $(0, \pm b e)$ and directrices are $y = \pm b / e$ with $e = 1 - a^{2} / b^{2}$ .

Q6. What is the equation of the chord of an ellipse joining two eccentric angles?

The chord joining $P (α)$ and $P (β)$ on $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ is $\frac{x}{a} cos \frac{α + β}{2} + \frac{y}{b} sin \frac{α + β}{2} = cos \frac{α - β}{2}$ . This form is very handy for focal-chord problems: setting the chord through $(\pm a e, 0)$ gives a compact eccentricity relation.

Q7. When does a general second-degree equation represent an ellipse?

The equation $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ represents an ellipse when the discriminant $Δ = ab c + 2 f g h - a f^{2} - b g^{2} - c h^{2}$ is non-zero and $h^{2} < ab$ . If additionally $a = b$ and $h = 0$ , it becomes a circle. If $Δ = 0$ , the conic degenerates (into a point or pair of lines).

Q8. How is the auxiliary circle used in problem solving?

The auxiliary circle $x^{2} + y^{2} = a^{2}$ lets you parametrise the ellipse cleanly as $(a cos θ, b sin θ)$ . Many locus and chord problems on the ellipse become easier if you first translate to the corresponding problem on the circle (by scaling $y$ by $a / b$ ) and then translate back at the end.

Q9. Is the ellipse important for JEE?

Yes. JEE Main routinely asks for standard parameters (foci, directrices, LR, eccentricity), focal chords, tangents and normals. JEE Advanced adds harder problems on parametric chords, conjugate diameters, director circle, and reflection properties.

Previous year questions on Introduction to Ellipse

17 questions from past papers, each with a step-by-step solution.

Show all 17 questions

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1. Definition of an Ellipse

2. General Equation of a Conic as an Ellipse

3. Standard Equation of an Ellipse (Horizontal, a>b)

3.1 All the standard parameters

3.2 Vertical case (b>a): major axis on the y-axis

4. Shifted Ellipse: Centre at (h,k)

5. Auxiliary Circle and the Eccentric Angle

6. Parametric Representation

6.1 Equation of the chord joining α and β

7. Focal Distance and the "Sum of Distances" Property

8. Position of a Point with Respect to an Ellipse

9. Area Enclosed by the Ellipse

Common Mistakes to Avoid

Frequently Asked Questions

Q1. What is an ellipse in the simplest terms?

Q2. How do I identify the major axis of an ellipse from its equation?

Q3. What is the length of the latus rectum and why is it useful?

Q4. What is the eccentric angle, and how is it different from the polar angle?

Q5. How do I quickly find foci and directrices from the standard equation?

Q6. What is the equation of the chord of an ellipse joining two eccentric angles?

Q7. When does a general second-degree equation represent an ellipse?

Q8. How is the auxiliary circle used in problem solving?

Q9. Is the ellipse important for JEE?

Previous year questions on Introduction to Ellipse

Ready to master Ellipse?

3. Standard Equation of an Ellipse (Horizontal, $a > b$ )

3.2 Vertical case ( $b > a$ ): major axis on the $y$ -axis

4. Shifted Ellipse: Centre at $(h, k)$

6.1 Equation of the chord joining $α$ and $β$