www.gradeup.co One of the most important topics from JEE point of view is the . Further and constitute of 2-3 questions every year. Read and revise all the important topics from ellipse and hyperbola. Download the pdf of the Short Notes on Ellipse and Hyperbola from the link given at the end of the article.

1.Basic of conic sections

Conic Section is the locus of a point which moves such that the ratio of its distance from a fixed point to its distance from a fixed line is always constant.

The focus is the fixed point, directrix is the fixed straight line, eccentricity is the constant ratio and axis is the line passing through the focus and perpendicular to the directrix.

The point of intersection with axis is called vertex of the conic.

1.1 General Equation of the conic

PS2 = e2 PM2

Simplifying above, we will get ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, this is the general equation of the conic.

1.2 Nature of the conic

The nature of the conic depends on eccentricity and also on the relative position of the fixed point and the fixed line.

Discriminant ‘Δ’ of a second-degree equation is defined as:

www.gradeup.co Nature of the conic depends on

Case 1: When the fixed point ‘S’ lies on the fixed line i.e., on directrix. Then the discriminant will be 0. Then the general equation of the conic will represent two lines

Case 2: When the fixed point ‘S’ does not lies on the fixed line i.e., not on directrix. Then the discriminant will not be 0. Then the general equation of the conic will represent , ellipse, and hyperbola.

(a) Parabola: When eccentricity is 1; h2 = ab

(b) Ellipse: When eccentricity is <1; h2

(c) Hyperbola: When eccentricity is >1; h2 > ab

When a+ b = 0 then it becomes rectangular hyperbola.

2. Ellipse

Ellipse is the locus of a point in a plane which moves such that the sum of its distances from two fixed points in the same plane is always constant i.e., | PF1 | + |PF2| = 2a

Simplifying above equation, the final equation of the ellipse will be,

where b2 = a2 - c2

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2.1 Nomenclature of the ellipse

Above equations are valid when axes are x and y-axis. In the case of axes different from x and y-axis, the equation will be

, and L1 = 0 and L2 = 0 are the major and minor axis.

2.2 Terminology of the ellipse (when axes are x and y axis)

(a) The line joining two focuses (F1 and F2) are called as the focal axis or major axis.

(b) Distance between F1 and F2 is called as focal length.

(c) The points at (a,0) and (-a,0) are the coordinates of the vertex.

(d) Length of the major axis is 2a and that of the minor axis is 2b

(e) A chord of ellipse perpendicular to the major axis is called as double ordinate.

(f) A chord which passes through the focus of the ellipse is called as focal chord

www.gradeup.co (g) double ordinate passing through focus or a focal chord perpendicular to the major axis and passes through focus is called as latus rectum

(h) Length of latus rectum = = 2a (1 – e2) = 2e (distance between focus and the directrix corresponding to the focus)

(i) Any chord of the ellipse passing through the centre (point of intersection of the major and minor axis) is bisected at this point and hence it is called as diameter.

2.3 Eccentricity

⇒ c = ae

For the ellipse, ,

⇒ b2 = a2 – c2 = a2 (1 – e2)

⇒

When a >b, then the equation of the ellipse will be and rest will follow accordingly. However

when a

For instance, Directrix now would be : y = ±b/e

Vertices: (0, ±b)

Length of latus rectum:2a2/b

Focus: (0, ±be) since c = be now

2.4 Auxiliary

The circle described on the major axis of the ellipse as the diameter is called an auxiliary circle.

www.gradeup.co P: (a cosθ, b sin θ)

Q: (a cosθ, a sin θ)

Equation of the circle is: x2 + y2 = a2

When a perpendicular is drawn from a point Q on the auxiliary circle to cut the ellipse at P then P, Q are called as corresponding points and θ is called as eccentric angle of the point P.

2.5 Parametric Representation of an ellipse

In the auxiliary topic, we saw that P is on the ellipse and thus it satisfies the equation of the ellipse.

Parameter here is θ.

This gives us the parametric representation of the ellipse with point P: (a cosθ, b sin θ)

2.6 Position of a point w.r.t to an ellipse

Let S:

Point P lies outside the circle,

www.gradeup.co S1 > 0 : Point is outside the curve

S1 < 0: Point is inside the curve

S1 = 0: Point is on the curve.

2.7 Line and Ellipse

L: y = mx + c

S:

A line can be a to the ellipse, it may cut the ellipse or it may not cut/touch the ellipse at all.

To find that, we will make a quadratic in ‘x’ using equations of L and S.

If discriminant D> 0 ⇒ Two roots ⇒ Intersect ⇒ Secant

If D= 0 ⇒ One root ⇒ Touching the ellipse ⇒Tangent

If D< 0 ⇒ No roots ⇒ Neither secant nor tangent

2.8 Condition of Tangency

D = 0

⇒

Thus the equation of the tangent would be

L: y = mx + c

2.9 Number of from a given point (h,k) to the ellipse

We have,

p (h,k)

Thus,

(k – mh)2 = a2m2 + b2 which is a quadratic in m.

This suggests that from a given point P(h,k) we can draw at max two tangents.

www.gradeup.co 2.10 Angle between the two tangents

Let the angle between the tangents be θ

From above quadratic equation, we know slope: m1 and m2

Thus,

From the quadratic equation in m,

2.11 Director circle of the ellipse

When θ = 90° then

m1m2 = -1 x2 + y2 = a2 + b2 : Equation of director circle.

Director circle is the locus of all those point from where the ellipse can be seen at angle 90°.

2.12 Equation of the chord of an ellipse joining α and β on it

2.13 The equation of Tangent, Normal and Chord of Contact

Tangents:

(a) Cartesian Tangent

(b)Slope form

(c) Parametric form

www.gradeup.co Normal:

(a) Cartesian Normal

(b) Slope form

(c) Parametric form

2.14 Chord of contact:

The equation of chord of contact will be like that of the tangent. Thus, a line when touches the ellipse will be tangent and the same line when cuts the ellipse will be the chord of contact.

Equation: T = 0 (Similar to that of tangent equation)

2.15 Pair of tangents:

Equation of pair of tangents would be

2 SS1 = T , where S is the equation of the ellipse, S1 is the equation when a point P (h, k) satisfies S, T is the equation of the tangent.

2.16 Equation of the chord whose middle point is (x1, y1):

T = S1

3 Hyperbola

Hyperbola is the locus of a point in a plane such that the difference of its distance from two fixed point in the same plane is always constant.

Here, e> 1 and h2 >ab

Let P (h, k)

Thus, according to the definition, equation would be

|PF1 | - |PF2| = 2a

Solving this condition will give the equation of hyperbola as:

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, where b2 = a2 (e2 – 1)

3.1 Nomenclature of hyperbola:

Eccentricity (e):

Foci: (c,0) and (-c,0) or (ae, 0) and (-ae,0)

Transverse axis (TA): The line segment containing the foci is known as the transverse axis and the length is 2a

Conjugate axis (CA): The line segment containing the points B1 and B2 is called as conjugate axis and the length is 2b.

Centre of the hyperbola: Point of the intersection of TA and CA is known as the center.

www.gradeup.co Both TA and CA together known as the principle axes of the hyperbola.

Vertices: A1 (a,0) and A2 (-a,0)

Focal axis: Line containing the fix points F1 and F2 (called as foci) is called as the focal axis

Focal length: The distance between F1 and F2 is called the focal length

Focal chord: A chord passing through a focus is called the focal chord.

Latus rectum (LR): LR is the focal chord which is perpendicular to TA and whose length is 2b2/a = 2a(e2 – 1)= 2e ( ae – a/e) = 2e (Distance between focus and corresponding foot of directrix)

Double ordinate: A chord which is perpendicular to the transverse axis.

Directrix: x = ±a/e

3.2 Conjugate Hyperbola

Corresponding to every hyperbola there exist a hyperbola such that the conjugate axis and the transverse axis of one is equal to the transverse axis and the conjugate axis of the other. Such are called as the conjugate hyperbola

If, hyperbola, H:

Then conjugate hyperbola, CH:

If e1 is the eccentricity of the hyperbola and e2 is the eccentricity of the conjugate hyperbola then

2 2 , whereas e1 = 1 + and e1 = 1 +

3.3 Rectangular Hyperbola

A hyperbola will be converted to a rectangular hyperbola if a = b i.e., TA = CA. This hyperbola is also known as an equilateral hyperbola.

Eccentricity, e =√2

Length of latus rectum= 2a = CA = TA

www.gradeup.co 3.4 Auxiliary Circle and Eccentric angle

The circle described on the transverse axis of the hyperbola as the diameter is called an auxiliary circle.

Thus, it’s equation would be x2 + y2 = a2

And for conjugate hyperbola, the equation would be x2 + y2 = b2

3.5 Parametric coordinates of a point on the hyperbola

P (a sec θ, b tan θ)

From point P, a perpendicular line is drawn to meet the TA at (let’s say).

From T a tangent is drawn on the auxiliary circle, meeting the circle at Q.

These two points: P and Q are called as the corresponding point and Q is called as the eccentric angle of the point P.

3.6 Position of a point with respect to the hyperbola

Point P lies outside the hyperbola,

S1 < 0 : Point is outside the curve

www.gradeup.co S1 > 0: Point is inside the curve

S1 = 0: Point is on the curve

3.7 Line and Parabola

Let, L: y = mx + c

S:

A line can be a tangent to the hyperbola, it may cut the hyperbola or it may not cut/touch the hyperbla at all.

To find that, we will make a quadratic in ‘x’ using equations of L and S.

If discriminant D> 0 ⇒ Two roots ⇒ Intersect ⇒ Secant

If D= 0 ⇒ One root ⇒ Touching the ellipse⇒ Tangent

If D< 0 ⇒ No roots ⇒ Neither secant nor tangent

3.8 Condition of Tangency

D = 0

⇒

Thus the equation of the tangent would be

L: y = mx + c

3.9 Number of tangents from a given point (h,k) to the hyperbola

We have,

P (h,k)

Thus,

(k – mh)2 =a2m2 - b2 which is a quadratic in m.

This suggests that from a given point P(h,k) we can draw at maximum two tangents.

www.gradeup.co 3.10 Angle between the two tangents

Let the angle between the tangents be θ

From above quadratic equation we know slope: m1 and m2

Thus,

From the quadratic equation in m,

3.11 Director circle of the ellipse

When θ = 90° then m1m2 = -1 x2 + y2 = a2 - b2 : Equation of director circle.

Now, director circle can be imaginary or real.

Real director circle: when the length of TA > length of CA

Imaginary director circle: when the length of TA < length of CA

Point circle: when the length of TA = length of CA ie., in case of a rectangular hyperbola.

Thus, director circle is the locus of all those point from where the ellipse can be seen at angle 90.

3.12 Equation of the chord of a hyperbola joining α and β on it

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3.13 Equation of Tangent, Normal and Chord of Contact

Tangents: a) Cartesian Tangent

b) Slope form

Parametric form

Normal: a) Cartesian Normal

b) Slope form

www.gradeup.co c) Parametric form

3.14 Chord of contact:

The equation of chord of contact will be similar to that of the tangent. Thus a line when touches the hyperbola will be tangent and the same line when cuts the hyperbola will be the chord of contact.

Equation: T = 0 (Similar to that of tangent equation)

3.15 Pair of tangents:

The equation of pair of tangents would be

2 SS1 = T , where S is the equation of the hyperbola, S1 is the equation when a point P (h,k) satisfies S, T is the equation of the tangent.

3.16 Equation of the chord whose middle point is (x1, y1):

T = S1

3.17 Rectangular Hyperbola

This is a special kind of hyperbola when the length of traverse and conjugate axis are equal.

The equation is x2 – y2 = a2

However, rotating the coordinate axis through an angle of 450, we get an another form of this rectangular hyperbola. i.e., XY = c2

3.18 Parametric coordinates of (RH)

(ct, c/t ) where t≠ 0 and t ∈ R

Two points on this hyperbola can be represented by (ct1,c/t1) and

(ct2, c/t2)

3.19 The equation of chord joining t1 and t2 points on the rectangular hyperbola

Two points on this hyperbola can be represented by (ct1,) and (ct2,)

Slope of the line joining t1 and t2 :

Thus the equation of chord:

www.gradeup.co x + t1t2 y = c (t1 + t2)

3.20 Equation of tangent of the rectangular hyperbola

a)Cartesian form at the point (x1,y1)

b) Parametric form

3.21 The equation of normal of the rectangular hyperbola

Parametric form:

(y -c/t) = t2 ( x – ct)

Equation of a chord whose middle point is (h,k)

T = S1

Thus, equation is

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