Introduction to Conic Section

Q: What is a conic section in simple terms?

A conic section is any curve obtained by slicing a double napped right circular cone with a flat plane. Depending on the tilt of the slicing plane you get a circle, an ellipse, a parabola or a hyperbola. Equivalently, every such curve is the locus of a point whose distance from a fixed point (the focus) and from a fixed line (the directrix) are in a constant ratio e.

Q: How is eccentricity related to the shape of a conic?

Eccentricity e measures how far a conic is from being circular. e = 0 gives a circle, 0 1 gives a hyperbola. A rectangular hyperbola has e = sqrt(2).

Q: What is the difference between a parabola and a hyperbola?

A parabola has eccentricity exactly 1 and has a single unbounded branch. A hyperbola has eccentricity greater than 1 and has two disconnected branches opening in opposite directions. A parabola comes from a plane parallel to one generator of the cone, while a hyperbola comes from a plane that cuts both nappes.

Q: How do you identify a conic from its general second-degree equation?

First compute the discriminant Delta = abc + 2fgh - af^2 - bg^2 - ch^2. If Delta = 0 the conic is degenerate. Only if Delta is non-zero do you compare h^2 with ab: h^2 = ab gives a parabola, h^2 ab gives a hyperbola.

Q: Why is a circle considered a special case of an ellipse?

As the eccentricity of an ellipse decreases, its two foci move closer together and its directrices move further away. In the limit e = 0 the foci merge into a single centre and the directrices recede to infinity, so the curve becomes a circle. Every circle is an ellipse, but not every ellipse is a circle.

Q: What are the axis and vertex of a conic?

The axis is the line of symmetry of the conic, passing through the focus and perpendicular to the directrix. A vertex is a point where the conic meets its axis. A parabola has one vertex, while an ellipse and a hyperbola have two each.

Q: Do conics appear in JEE Main and JEE Advanced?

Yes. Conic sections form a major part of Coordinate Geometry in both JEE Main and JEE Advanced. Parabola, ellipse and hyperbola each have their own dedicated chapters, and this introductory classification is the foundation for all of them.

MathsParabolaFor JEE aspirants

A conic section is the locus of a point that moves in a plane so that the ratio of its distance from a fixed point (called the focus) to its perpendicular distance from a fixed straight line (called the directrix) is a constant. This constant ratio is the eccentricity $e$ . Depending on the value of $e$ , one single definition generates four different curves: a parabola ( $e = 1$ ), an ellipse ( $0 < e < 1$ ), a hyperbola ( $e > 1$ ), and a circle ( $e = 0$ , a limiting case). The same four curves are obtained by slicing a double napped right circular cone with a plane at different angles, which is why they are called conic sections.

Key Formulas - Quick Reference

Focus-directrix definition: $\frac{P S}{P M} = e$ , where $S$ is the focus and $M$ is the foot of the perpendicular from $P$ on the directrix.
Eccentricity classification: $e = 0 \Rightarrow$ circle; $0 < e < 1 \Rightarrow$ ellipse; $e = 1 \Rightarrow$ parabola; $e > 1 \Rightarrow$ hyperbola.
General focus-directrix equation: if $S = (p, q)$ and the directrix is $ℓ x + m y + n = 0$ , then $(ℓ^{2} + m^{2}) [(x - p)^{2} + (y - q)^{2}] = e^{2} (ℓ x + m y + n)^{2}$ .
General second-degree form: every conic can be written as $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ for suitable constants.
Discriminant: $Δ = ab c + 2 f g h - a f^{2} - b g^{2} - c h^{2}$ , which is the determinant of the symmetric matrix of coefficients. The conic is degenerate exactly when $Δ = 0$ .
Nature test (only after checking $Δ \neq = 0$ ): $h^{2} = ab \Rightarrow$ parabola; $h^{2} < ab \Rightarrow$ ellipse; $h^{2} > ab \Rightarrow$ hyperbola; and $h^{2} > ab$ together with $a + b = 0 \Rightarrow$ rectangular hyperbola.
Cone-angle test: with $α$ the semi-vertical angle of the cone and $β$ the angle between the cutting plane and the axis: $β = 9 0^{\circ} \Rightarrow$ circle; $α < β < 9 0^{\circ} \Rightarrow$ ellipse; $β = α \Rightarrow$ parabola; $β < α \Rightarrow$ hyperbola.

1. What is a Conic Section?

A conic section, or simply a conic, is defined as the locus of a point $P$ that moves in a plane such that its distance from a fixed point bears a constant ratio to its perpendicular distance from a fixed straight line.

Formal definition: A conic is the locus of a point $P$ such that

$\frac{distance of P from the fixed point}{perpendicular distance of P from the fixed line} = e (a constant) .$

Three objects appear in this definition:

Focus ( $S$ ): the fixed point.
Directrix: the fixed straight line.
Eccentricity ( $e$ ): the constant ratio $\frac{P S}{P M}$ , where $P M$ is the perpendicular distance from $P$ to the directrix.

Figure 1: A conic is the locus of a point

P

whose distance

P S

from the focus and perpendicular distance

P M

from the directrix stay in the fixed ratio

e

. The axis is the line through

S

perpendicular to the directrix, and

Z

is its foot on the directrix.

Standard terminology

These words are used throughout the parabola, ellipse and hyperbola chapters, so it pays to fix them now.

Term	Meaning
Axis	The line through the focus, drawn perpendicular to the directrix. The conic is symmetric about this line.
Vertex	A point where the conic meets its axis. A parabola has one vertex; an ellipse and a hyperbola have two.
Centre	The point that bisects every chord passing through it. An ellipse and a hyperbola have a centre; a parabola does not.
Focal chord	Any chord of the conic that passes through a focus.
Double ordinate	Any chord drawn perpendicular to the axis.
Latus rectum	The focal chord that is perpendicular to the axis, that is, the double ordinate through the focus.

2. Deriving the Equation of a Conic

Suppose the focus is $S = (p, q)$ and the directrix is the line $ℓ x + m y + n = 0$ . Let $P = (x, y)$ be any point on the conic. Then

P S = (x - p)^{2} + (y - q)^{2}, P M = \frac{∣ ℓ x + m y + n ∣}{ℓ ^{2} + m ^{2}} .

The focus-directrix definition gives $\frac{P S}{P M} = e$ . Squaring both sides removes the modulus and the square root:

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

Both sides are quadratic in $x$ and $y$ , so on expanding and collecting terms this always reduces to

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

which is the general second-degree equation. Every conic, no matter how tilted or shifted, can be written in this form.

JEE AdvancedFive points determine a conic. Dividing the general equation by any one non-zero coefficient leaves

5

independent constants, so five points in general position determine a unique conic through them. This is why conic problems so often supply exactly five conditions.

3. Sections of a Double Napped Cone

The name "conic section" is literal: each of these curves is the shape you get when a double napped right circular cone is sliced by a plane. Two nappes are needed because a hyperbola has two branches, one on each nappe.

Figure 2: A double napped right circular cone. The two nappes meet at the vertex

V

, the axis is the line of symmetry, a generator is any straight line lying on the surface through

V

, and

α

is the semi-vertical angle between the axis and a generator.

Let $α$ be the semi-vertical angle of the cone and let $β$ be the angle between the cutting plane and the axis. The section obtained depends only on how $β$ compares with $α$ .

Angle of the cutting plane	Section obtained	What the plane does
$β = 9 0^{\circ}$	Circle	Plane perpendicular to the axis, cutting one nappe
$α < β < 9 0^{\circ}$	Ellipse	Plane tilted, still cutting right through one nappe
$β = α$	Parabola	Plane parallel to exactly one generator
$0^{\circ} \leq β < α$	Hyperbola	Plane steep enough to cut both nappes

Figure 3: The four non-degenerate conic sections. The shaded parallelogram is the cutting plane and the bold curve is the section; dashed portions of the curve lie on the far side of the cone.

Reading the same four cases in words:

Plane perpendicular to the axis (parallel to the base of a nappe): the section is a circle.
Plane inclined to the axis at an angle greater than $α$ but not perpendicular to it: the plane cuts clean through one nappe and the section closes up into an ellipse.
Plane parallel to exactly one generator ( $β = α$ ): the section can never close, and it is a parabola.
Plane inclined at an angle less than $α$ , so that it meets both nappes: the section is a hyperbola, with one branch on each nappe.

When the plane passes through the vertex

All four cases above assume the cutting plane misses the vertex. If the plane passes through the vertex, the section collapses into a degenerate conic.

Figure 4: The three degenerate sections. Passing through the vertex, a steep plane meets the cone along two generators, a tangent plane meets it along one generator counted twice, and a shallow plane meets it only at the vertex.

$β < α$ : the plane meets the surface along two generators, giving a pair of intersecting straight lines.
$β = α$ : the plane is tangent along a single generator, giving a pair of coincident straight lines.
$β > α$ : the plane touches the cone only at the vertex, so the real locus is a single point.

Note: The circle, parabola, ellipse and hyperbola are the non-degenerate conics. A pair of lines, a single point and the empty set are the degenerate conics; they arise only when the cutting plane passes through the vertex.

4. The Four Conics and Their Eccentricity

Every non-degenerate conic can be built from a focus and a directrix. Raising $e$ opens the curve out: a small $e$ gives a nearly circular closed curve, $e = 1$ is the exact borderline at which the curve stops closing, and $e > 1$ splits it into two branches.

Figure 5: The four conics, each drawn directly from the focus-directrix definition. Every orange curve is the exact locus of points satisfying

P S = e \cdot P M

for the value of

e

shown; the panels are scaled independently so that each curve fills its own frame.

Why the circle is the odd one out: putting

e = 0

P S = e \cdot P M

would force

P S = 0

, so a circle cannot literally be produced from a focus and a directrix at a finite distance. The correct statement is that a circle is the limiting case in which the two foci of an ellipse merge into the centre while the directrix recedes to infinity.

5. Distinguishing Between Conics

Given the general second-degree equation $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ , we identify the conic using two quantities, in this order:

Δ = ab c + 2 f g h - a f^{2} - b g^{2} - c h^{2} = ah g hb f g f c, then compare h^{2} with ab .

Case I - The conic is degenerate ( $Δ = 0$ )

In the focus-directrix picture this is exactly the situation in which the focus lies on the directrix. The locus then breaks up into straight lines through $S$ , and $h^{2}$ versus $ab$ tells you which kind:

Eccentricity	Coefficient condition	Real locus
$e > 1$	$h^{2} > ab$	Two real distinct lines intersecting at $S$
$e = 1$	$h^{2} = ab$	A pair of coincident lines through $S$
$e < 1$	$h^{2} < ab$	Two imaginary lines meeting at the real point $S$ , so the real locus is just the single point $S$

Careful: in the last row the locus is not empty. Take the directrix as the

y

-axis through

S

at the origin. Then

P S = e P M

becomes

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

, that is

y^{2} + (1 - e^{2}) x^{2} = 0

. For

e < 1

both terms are non-negative, so the only real solution is

x = y = 0

, the point

S

itself.

Case II - The conic is non-degenerate ( $Δ \neq = 0$ )

Now the curve is a genuine conic, and its type is fixed by the $h^{2}$ versus $ab$ test:

Conic	Eccentricity	Test on coefficients
Circle	$e = 0$	$Δ \neq = 0, h = 0, a = b$
Parabola	$e = 1$	$Δ \neq = 0, h^{2} = ab$
Ellipse	$0 < e < 1$	$Δ \neq = 0, h^{2} < ab$
Hyperbola	$e > 1$	$Δ \neq = 0, h^{2} > ab$
Rectangular hyperbola	$e = 2$	$Δ \neq = 0, h^{2} > ab, a + b = 0$

Quick sanity check: if

h = 0

(no

x y

term) and

Δ \neq = 0

, just compare

a

and

b

. Same sign and equal gives a circle; same sign but unequal gives an ellipse; opposite signs give a hyperbola; exactly one of them zero gives a parabola.

JEE AdvancedCentre of a conic. Solving

a x + h y + g = 0

and

h x + b y + f = 0

simultaneously gives the centre

(\frac{h f - b g}{ab - h ^{2}}, \frac{g h - a f}{ab - h ^{2}}) .

The denominator

ab - h^{2}

vanishes precisely when

h^{2} = ab

, which is the parabola condition. That is the algebraic reason a parabola has no centre. Also,

a + b = 0

makes the asymptotes of a hyperbola perpendicular, which is exactly what "rectangular" means, and forces

e = 2

Solved Examples

Solved Example 1

Find the equation of the conic whose focus is

(- 1, - 2)

, directrix is

x - 2 y + 3 = 0

, and eccentricity is

1

(that is, a parabola).

Solution:

Let $P (x, y)$ be any point on the conic. Since $e = 1$ , the definition gives $S P = P M$ , so $S P^{2} = P M^{2}$ :

(x + 1)^{2} + (y + 2)^{2} = (\frac{x - 2 y + 3}{1 + 4})^{2} .

Multiplying through by $5$ and expanding both sides:

$5 [(x + 1)^{2} + (y + 2)^{2}] = (x - 2 y + 3)^{2}$

$5 (x^{2} + y^{2} + 2 x + 4 y + 5) = x^{2} + 4 y^{2} + 9 - 4 x y + 6 x - 12 y$

4 x^{2} + 4 x y + y^{2} + 4 x + 32 y + 16 = 0

As a check, $a = 4, b = 1, h = 2$ , so $h^{2} = 4 = ab$ , which confirms a parabola.

Solved Example 2

Identify the conic represented by

3 x^{2} - 2 x y + 3 y^{2} + 4 x - 6 y + 5 = 0

Solution:

Comparing with $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ :

$a = 3, b = 3, 2 h = - 2 \Rightarrow h = - 1, 2 g = 4 \Rightarrow g = 2, 2 f = - 6 \Rightarrow f = - 3, c = 5$ .

Step 1 - Check $Δ$ :

Δ = ab c + 2 f g h - a f^{2} - b g^{2} - c h^{2} = (3) (3) (5) + 2 (- 3) (2) (- 1) - 3 (9) - 3 (4) - 5 (1)

= 45 + 12 - 27 - 12 - 5 = 13 \neq = 0,

so the conic is non-degenerate.

Step 2 - Compare $h^{2}$ with $ab$ :

$h^{2} = 1$ and $ab = 9$ . Since $h^{2} < ab$ with $Δ \neq = 0$ , the conic is an ellipse. It is not a circle, because $h \neq = 0$ .

Solved Example 3

Find the equation of the parabola whose focus is

(0, 0)

and whose directrix is

3 x - 4 y + 2 = 0

Solution:

Here $e = 1$ , so $S P^{2} = P M^{2}$ gives

x^{2} + y^{2} = (\frac{3 x - 4 y + 2}{9 + 16})^{2} = \frac{( 3 x - 4 y + 2 ) ^{2}}{25} .

Multiplying by $25$ :

$25 (x^{2} + y^{2}) = (3 x - 4 y + 2)^{2} = 9 x^{2} + 16 y^{2} + 4 - 24 x y + 12 x - 16 y$

16 x^{2} + 24 x y + 9 y^{2} - 12 x + 16 y - 4 = 0

Check: $a = 16, b = 9, h = 12$ , and $h^{2} = 144 = ab$ , as a parabola requires.

Solved Example 4

Identify the conic represented by

x^{2} - 5 x y + 4 y^{2} + x + 2 y - 3 = 0

Solution:

Reading off the coefficients: $a = 1, b = 4, 2 h = - 5 \Rightarrow h = - \frac{5}{2}, 2 g = 1 \Rightarrow g = \frac{1}{2}, 2 f = 2 \Rightarrow f = 1, c = - 3$ .

Step 1 - Check $Δ$ :

Δ = (1) (4) (- 3) + 2 (1) (\frac{1}{2}) (- \frac{5}{2}) - (1) (1)^{2} - (4) (\frac{1}{2})^{2} - (- 3) (\frac{25}{4})

= - 12 - \frac{5}{2} - 1 - 1 + \frac{75}{4} = \frac{9}{4} \neq = 0,

so the conic is non-degenerate.

Step 2 - Compare $h^{2}$ with $ab$ :

$h^{2} = \frac{25}{4} = 6.25$ and $ab = 4$ . Since $h^{2} > ab$ , the conic is a hyperbola. Finally $a + b = 5 \neq = 0$ , so it is not a rectangular hyperbola.

Solved Example 5

Here

h^{2} > ab

, which looks like a hyperbola. Show that

x^{2} - 5 x y + 4 y^{2} + x + 2 y - 2 = 0

is in fact not a hyperbola.

Solution:

The coefficients are the same as in Example 4 except that $c = - 2$ . So $h^{2} = \frac{25}{4} > 4 = ab$ still holds, which tempts you to write "hyperbola" straight away. Check $Δ$ first:

Δ = (1) (4) (- 2) + 2 (1) (\frac{1}{2}) (- \frac{5}{2}) - (1) (1)^{2} - (4) (\frac{1}{2})^{2} - (- 2) (\frac{25}{4})

= - 8 - \frac{5}{2} - 1 - 1 + \frac{25}{2} = 0.

Since $Δ = 0$ , the conic is degenerate, and because $h^{2} > ab$ it must be a pair of real distinct lines. Factorising confirms this:

x^{2} - 5 x y + 4 y^{2} + x + 2 y - 2 = (x - y - 1) (x - 4 y + 2) .

So the locus is the pair of straight lines $x - y - 1 = 0$ and $x - 4 y + 2 = 0$ , which meet at $(2, 1)$ . This is precisely why $Δ$ must always be tested before the $h^{2}$ versus $ab$ comparison.

Common Mistakes to Avoid

Watch out

Forgetting to check $Δ$ first. The $h^{2}$ versus $ab$ test only classifies the conic after you have confirmed $Δ \neq = 0$ . Solved Example 5 is exactly this trap: the equation passes the "hyperbola" test but is really a pair of straight lines.
Confusing "focus" and "vertex". The focus is the fixed reference point in the definition; the vertex is where the conic actually meets its axis. For a parabola the two are different points.
Losing the modulus in $P M$ . $P M = \frac{∣ ℓ x + m y + n ∣}{ℓ ^{2} + m ^{2}}$ . Squaring makes the modulus disappear, but you must divide by $ℓ^{2} + m^{2}$ and not by $ℓ^{2} + m^{2}$ when you clear the fraction.
Treating $e = 0$ as an ordinary ellipse. A circle is the limiting case in which the foci merge at the centre and the directrix goes off to infinity. It is not a "very flat" ellipse; it is the roundest one.
Reading $h^{2} = ab$ as "parabola" when $Δ = 0$ . If both $Δ = 0$ and $h^{2} = ab$ , the equation represents a pair of coincident lines, not a parabola.
Forgetting the second nappe. A hyperbola needs a double cone. If you picture only one nappe, you will never see the second branch, and you may wrongly call a steep cut a parabola.

Frequently Asked Questions

Q1. What is a conic section in simple terms?

A conic section is any curve you get by slicing a double napped right circular cone with a flat plane. Depending on the tilt of the slicing plane you get a circle, an ellipse, a parabola or a hyperbola. Equivalently, every such curve is the locus of a point whose distance from a fixed point (the focus) and from a fixed line (the directrix) are in a constant ratio $e$ .

Q2. How is eccentricity related to the shape of a conic?

Eccentricity $e$ measures how far a conic is from being circular. $e = 0$ gives a circle, $0 < e < 1$ gives an ellipse (nearly circular for small $e$ and more elongated as $e \to 1$ ), $e = 1$ gives a parabola, which is the borderline case, and $e > 1$ gives a hyperbola. A rectangular hyperbola has $e = 2$ .

Q3. What is the difference between a parabola and a hyperbola?

A parabola has eccentricity exactly $1$ and consists of a single unbounded branch. A hyperbola has eccentricity greater than $1$ and consists of two disconnected branches opening in opposite directions. In the cone picture, a parabola comes from a plane parallel to one generator, while a hyperbola comes from a plane steep enough to cut both nappes.

Q4. How do you identify a conic from its general second-degree equation?

First compute $Δ = ab c + 2 f g h - a f^{2} - b g^{2} - c h^{2}$ . If $Δ = 0$ the conic is degenerate, meaning a pair of lines, a single point, or no real locus. Only if $Δ \neq = 0$ do you compare $h^{2}$ with $ab$ : $h^{2} = ab$ gives a parabola, $h^{2} < ab$ gives an ellipse (a circle when $h = 0$ and $a = b$ ), and $h^{2} > ab$ gives a hyperbola.

Q5. What is a degenerate conic?

A degenerate conic is what you get when the cutting plane passes through the vertex of the cone. Instead of a proper curve you get a pair of intersecting lines, a pair of coincident lines, or a single point. Algebraically this happens exactly when $Δ = 0$ in the general second-degree equation, and in the focus-directrix picture it happens exactly when the focus lies on the directrix.

Q6. Why is a circle considered a special case of an ellipse?

An ellipse has two foci and two directrices. As $e$ decreases the two foci move closer together and the directrices move further out; in the limit $e = 0$ the foci merge into a single centre and the directrices recede to infinity, and the curve becomes a circle. So every circle is an ellipse, but not every ellipse is a circle. In the cone picture, the circle is the special ellipse obtained when the plane is exactly perpendicular to the axis.

Q7. What are the axis and vertex of a conic?

The axis is the straight line passing through the focus and perpendicular to the directrix; it is the line of symmetry of the conic. A vertex is a point where the conic meets its axis. A parabola has one vertex, while an ellipse and a hyperbola each have two vertices along the major or transverse axis.

Q8. Do conics appear in JEE Main and JEE Advanced?

Yes. Conic sections form a major part of the Coordinate Geometry portion of both JEE Main and JEE Advanced. Parabola, ellipse and hyperbola each get their own dedicated chapters, and questions frequently mix conic properties with straight lines, circles and calculus. This introductory classification is the foundation for all of them.

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Practice questions →Take a mock test

1. What is a Conic Section?

Standard terminology

2. Deriving the Equation of a Conic

3. Sections of a Double Napped Cone

When the plane passes through the vertex

4. The Four Conics and Their Eccentricity

5. Distinguishing Between Conics

Case I - The conic is degenerate (Δ=0)

Case II - The conic is non-degenerate (Δ=0)

Solved Examples

Common Mistakes to Avoid

Frequently Asked Questions

Q1. What is a conic section in simple terms?

Q2. How is eccentricity related to the shape of a conic?

Q3. What is the difference between a parabola and a hyperbola?

Q4. How do you identify a conic from its general second-degree equation?

Q5. What is a degenerate conic?

Q6. Why is a circle considered a special case of an ellipse?

Q7. What are the axis and vertex of a conic?

Q8. Do conics appear in JEE Main and JEE Advanced?

Ready to master Parabola?

Case I - The conic is degenerate ( $Δ = 0$ )

Case II - The conic is non-degenerate ( $Δ \neq = 0$ )