CBSE Class 11 Maths Revision Notes Chapter 11

Class 11 Mathematics  Revision Notes for Chapter 11 Conic Sections 

To understand Conic Sections, students can refer to the Class 11 Mathematics  Chapter 11 Notes provided by Extramarks to quickly recall equations and prepare this chapter for exams. Using revision notes is extremely helpful when there is less time to revise the chapter. The chapter’s content is organised by the subject matter experts of Extramarks in Class 11 Chapter 11 Mathematics Notes to gain conceptual clarity on the chapter. 

Access Class 11 Mathematics Chapter 11  Conic Section 

1. Conic Sections:

The locus of a point in a plane moves so that its distance from a fixed point to its perpendicular distance from a fixed straight line remains in a constant ratio. 

• Directrix refers to the constant straight line.
• Focus refers to the fixed point.
• The constant ratio is denoted by e as it is called eccentricity. 
• The line passing through the focus & perpendicular to the directrix is known as the axis.
• A vertex is the point of intersection between an axis and a conic.

2. Section of Right Circular Cone by Different Planes:

The section of a right circular cone by a plane passing through its vertex is a pair of straight lines that pass through its vertex. A circle is the section of a right circular cone by a plane parallel to its base. A parabola is the section of a right circular cone by a plane parallel to a generator of the cone. An ellipse or hyperbola is the section of a right circular cone by a plane that is neither parallel to any generator of the cone nor parallel/perpendicular to the axis of the cone.

3. General Equation of a Conic: Focal Directrix Property 

The general equation of conic if the focus of conic is (p,q) and the equation of directrix is lx+my+n=0 is: 

(l2=m2)[(x−p)2+(y−q)2)]=e2(lx+my+n)2≡ax2+2hxy+by2+2gx+2fy=c=0

4. Distinguishing Various Conics: 

The positions of the focus, the directrix, and the value of eccentricity affect the nature of the conic section. So, there are the following two cases: 

Case 1: When the focus lies on the directrix

In this case, 

Δ≡ abc + 2fgh −− af2 −− bg2 −− ch2 = 0Δ≡ abc + 2fgh −− af2 −− bg2 −− ch2 = 0 

A pair of straight lines are represented by the general equation of a conic in the following conditions: 

Real and distinct lines intersecting at focus if: e > 1 ≡ h2 > abe > 1 ≡ h2 > ab 

Coincident lines if: e = 1 ≡ h2 ⩾ abe = 1 ≡ h2 ⩾ ab 

Imaginary lines if: e < 1 ≡ h2 < abe < 1 ≡ h2 < ab 

Case 2: When the focus does not lie on the directrix, it represents any one of the following:

• a parabola, 
• an ellipse, 
• a hyperbola, or 
• a rectangular hyperbola 

Parabola

Definition and Terminology 

A parabola is a point’s location whose distance from a fixed point, or focus, equals its perpendicular distance from a fixed straight line, or directrix.

Parametric Representation: 

(at2,2at) represents the coordinates of a point on the parabola:

y2= 4ax 

i.e. the equations 

x=at2,y=2at together represent the parabola with t being the parameter.  The equation of a chord joining t1 & t2 is 

2x−−(t1+ t2)y + 2at1t2= 0. 

Pair of Tangents 

The equation to the pair of tangents drawn from any point (x1,y1) to the parabola y2=4ax is given by: SS1=T2 where: 

S≡y2−4ax,S1=y21−4ax1,T≡yy1−2a(x+x1)

Director Circle 

The director circle is the locus of the point of intersection of the perpendicular tangents to a curve. The equation for the parabola y2=4ax is x+a=0. This is the directrix of the parabola.

Chord of Contact 

The equation to the chord of contact of tangents that can be drawn from a point P(x1,y1) is yy1=2a(x+x1); (i.e., T=0)

Chord With a Given Middle Point 

y−y1=(2a/y1)(x−x1)≡T=S1  is the equation of the chord of a parabola y2=4ax whose middle point is (x1,y1) 

Ellipse

The standard equation of an ellipse is:

x2/a2+y2/b2=1, where a>b,b2=a2(1−e2) 

Auxiliary Circle/Eccentric Angle 

An auxiliary circle is a circle described on the major axis as the diameter. Let Q be a point on the auxiliary circle x2+y2=a2 such that QP produced will be perpendicular to the x-axis, then P & Q are known as the corresponding points on the ellipse and the auxiliary circle respectively. 

Parametric Representation 

x=a cosθ & y=b sinθ together represent the ellipse x2/a2+y2/b2=1 where θ is a parameter. If P(θ)=P(θ)= (a cosθ,bsinθ) is on the ellipse, then Q(θ)=(a cosθ, a sinθ) is on the auxiliary circle. Equation for the elliptical chord connecting two points with eccentric angles is provided by:

xacosa+?2+ybsina+?2=cosa-?2

Position of a Point With Respect to Ellipse: 

The point p(x1,y1) can lie either outside, inside or on the ellipse: 

[(x21/a2 )+ (y21/b2)]−1>0 (outside) 

[(x21/a2 )+ (y21/b2)]−1<0 (inside) 

x2/a2+y2/b2= 1= 0 (on)

Line and an Ellipse: 

The line y=mx+c meets the ellipse in two points real, coincident or imaginary according as c2 is <, = & > a2m2+b2 Hence, the tangent to the ellipse x2/a2+y2/b2=1, if c2=a2m2 + b2 is y=mx+c.

Tangents 

(a) Slope form y=mx ± √a2m2+b2 is tangent to the ellipse for m 

(b) Point form xx1/a2+yy1/b2=1 is tangent to the ellipse at point (x1,y1)

(c) Parametric for xcosθ/a+ysinθ/b=1 is a tangent to the ellipse at the point (a cosθ, b sinθ)(a cosθ, b sin⁡θ).

Director Circle: 

The director circle is the location of the intersection of the tangents that meet at right angles. This locus’ equation is x2+y2=a2+b2.

Diameter (Not in the syllabus)

Diameter is the locus of the middle points of a system of parallel chords with slope ‘m’ of an ellipse, which is a straight line passing through the centre of the ellipse.

Important Highlights: 

By referring to the ellipse x2/a2+y2/b2=1, it is clear that:

(a) If P be any point on the ellipse with S & S’ as foci, then the equation SP+SP′=2a.

(b) The external and internal angles between the focal distances of P are bisected by the tangent and normal at a point P on the ellipse. This is a well-known reflection property of the ellipse that states that the rays from one focus are reflected through another focus and vice-versa. Hence, the straight lines that join each focus to the foot of the perpendicular from the other focus upon the tangent at any point P meet on the normal PC.

Hyperbola  

It is a conic whose eccentricity is greater than unity (e>1)(e>1)

Standard Equation and Definitions: 

x2/a2+y2/b2=1 is the standard equation of the hyperbola, where b2=a2(e2−1)

Rectangular or Equilateral Hyperbola: 

The kind of hyperbola in which the lengths of the transverse and conjugate axis are equal is an equilateral hyperbola. 

Conjugate Hyperbola:

When the transverse & conjugate axes of one hyperbola are respectively the conjugate and the transverse axes of the other, they are conjugate hyperbolas of each other.

E.g. x2/a2 − y2/b2 = 1 & −x2/a2 +y2/b2 = 1 are conjugate hyperbolas of one another.

Auxiliary Circle:

An auxiliary circle is a circle drawn with centre C and T.A. as a diameter.

Parametric Representation:

x=asecθ & y=btanθ represents the hyperbola x2/a2+y2/b2=1 where the parameter is θ. If P(θ)=(asecθ,btanθ) is on the hyperbola, then Q(θ)=(acosθ, a tanθ) will be on the auxiliary circle. The following is the equation for the chord of the hyperbola connecting two locations with eccentric angles:

xacos?-?2–ybsin?+?2=cos?+?2

Position of a Point With Respect to Hyperbola:

The quantity S1= [(x21/a2 )+ (y21/b2)]−1 is positive, zero or negative depending upon whether the point (x1,y1) lies inside, on or outside the curve.

Line and a Hyperbola:

The straight line y=mx+c is a secant, a tangent that passes the outside the hyperbola x2/a2+y2/b2=1 according as c2 > or < a2m2−b2, respectively. 

Tangents:

(i) Slope Form: y=mx ± a2m2−b2 is the tangent to the hyperbola x2/a2+y2/b2=1

Director Circle:

The locus of the intersection point of tangents at right angles of each other is the director circle. 

Diameter (Not in the syllabus):

Diameter is the locus of the middle points of a system of parallel chords with slope ‘m’ of a hyperbola. It is a straight line that passes through the hyperbola’s centre and has the equation:

y=b2/a2m × x

Asymptotes (Not in the syllabus):

As the point on the hyperbola moves to infinity along the hyperbola, if the length of the perpendicular let fall from a point on a hyperbola to a straight line ends to zero, then the straight line is the asymptote.

Important Highlights:

1. A right angle at the corresponding focus is suspended by the portion of the tangent between the point of contact and the directrix. 
2. The angle between the focal radii is bisected by the tangent and normal at any point of a hyperbola.
3. Just as the corresponding directrix and the common points of intersection lie on the auxiliary circle, perpendicular from the foci on either asymptote meet it in the same points.

Circle:

A circle is the locus of a point whose separation from a fixed point (referred to as the centre) is constant (called radius).

1. Equation of Circle in Various Forms:

Some are: 

(a) The equation of a circle with the centre as origin & radius ‘r’ is x2+y2=r2

(b) The equation of a circle with centre (h,k)& radius ‘r’ is (x−h)2+(y−k)2=r2

2. Intercepts Made by Circle on the Axes:

The intercepts made by the circle x2+y2+2gx+2fy+c=0 are 2√g2−c and  2√f2−c respectively. If:

• The x-axis is divided into two independent places by the g2-c>0 circle.
• The circle at g2-c=0 touches the x-axis
• The g2-c<0 circles is entirely above or entirely below the x-axis.

3. Parametric Equation of the Circle:

The parametric equations of (x−h)2+(y−k)2=r2 are : x=h+rcosθ,y=k+rsinθ, −π<θ⩽π, where(h,k) is the centre, θ is a parameter and r is the radius.

4. Position of a Point with Respect to a Circle:

Depending on whether the point (x1,y1) is inside, on or outside the circle S ≡ x12+ y21+ 2gx + 2fy + c = 0, S1≡ x12+ y21 + 2xx1+ 2yy1+ c is < or = or > 0

5. Line and a Circle

Assuming L=0 be a line & S=0 be a circle, if r is the radius & p is the length of the perpendicular, then: 

(i) p>r⇔p>r⇔ the line does not meet or passes outside the circle. 

(ii) p=r⇔p=r⇔ the line touches the circle and is its tangent. 

(iii) p<r⇔p<r⇔ the line is a secant of the circle. 

1. iv) p=0⇒p=0⇒ the line is the diameter of the circle.

      6. Tangent:

Since the slope form: y=mx+cy is always a tangent to the circle x3+y2 = a2 if c2=a2(1+m2), the equation of tangent is y= mx ± a √(1+m2).

7. Family of Circles:

It is one of the important topics in coordinate geometry that refers to a large collection of circles.

8. Normal:

A line is normal/orthogonal to a circle and must pass through the centre of the circle.

9. Pair of Tangents From a Point:

SS1=T2 is the equation of a pair of tangents drawn from point A(x1,y1) to the circle x2+y2+2gx+2fy+c=0.

10. Length of Tangent and Power of a Point:

The length of a tangent from an external point (x1,y1) to circle S=x2+y2+2gx+2fy+c=0 is find out by L= √(x21+y21+2gx1+2f1y+c) = √S1

11. Director Circle:

It is the locus of the point of intersection of two perpendicular tangents.

12. Chord of Contact:

If two tangents are drawn from the point P(x1,y1) to the circle S=x2+y2+2gx+2fy+c=0, then xx1+yy1+g(x+x1)+f(y+y1)+c=0, T=0 will be the equation of the chord of contact T1T2 

13. Pole and Polar (Not in the syllabus):

If through a point P in a circle any straight line is drawn to meet the circle in Q and R, the locus of the point of intersection of the tangents at Q and R is polar of point P and P is the pole of the polar.

14. Equation of Chord with a Given Middle Point:

The equation of the chord of the circle in terms of its midpoint (x1,y1) is xx1+yy1+g(x+x1)+f(y+y1)+c=x21 + y21+2gx1+2fy1+c (designated by T=S1)

15. Equation of the Chord Joining Two Points of the Circle:

The equation of a straight line joining points α and β on the circle x2+y2 = a2 is

xcos [(a+β)/2] + ysin [(α+β)/2] = cos[(a−β)/2]

16. Common Tangents to the Two Circle:

The direct common tangents meet at a point that divides the line joining the centre of circles externally in the ratio of their radii.

17. Orthogonality of Two Circles:

Two circles S1=0 & S2=0 are said to be orthogonal if tangents at their point of intersection meet at a right angle. The condition for this is: 2g1g2+2f1f2=c1+c2

     18. Radical Axis and Radical Centre:

The radical axis is the locus of points whose powers w.r.t. the two circles are equal. The radical centre is the common point of intersection of the radical axes of three circles taken two at a time.

Revision Notes For CBSE Class 11 Mathematics Chapter 11

Conic Sections: Class 11 Mathematics Chapter 11 Revision Notes Summary 

This chapter is different from the ones previously studied in coordinate geometry. Students will learn higher-level concepts and complex representations of conic sections, along with the features of those sections and how they are represented on a 2D plane.