Parabola - Conic Section (Lecture-01)

A conic section or conic is the locus of a point P which moves in a plane such that the ratio of its distance from fixed point to the distance form the fixed line always remain same (constant).

PSPM=e

  • The fixed point known as Focus.
  • The fixed straight line is known as Directrix,
  • The constant ratio is called the Eccentricity denoted by e.
  • The line passing through the focus and perpendicular to the directrix is known as axis.
  • A point of intersection of a conic with its axis is called a vertex.
  • The point which bisects every chord of the conic passing through it, is called the centre of the conic.
  • The chord passing through the focus and perpendicular to the axis is known as Latus Rectum.

GENERAL EQUATION OF A CONIC SECTION:

Let coordinates of focus be S(p,q), equation of directrix be ax+by+c=0 and e be the eccentricity of a conic. Let P(h,k) be a moving point, then by definition.

SP=ePM

(hp)2+(kq)2=e|ah+bk+ca2+b2|

(hp)2+(kq)2=e(ah+bk+ca2+b2)2

Thus locus of (h,k) is

(xp)2+(yq)2=e2(ax+by+c)2(a2+b2)

This is equation of conic section which, when simplified, can be written in the form ax2+2hxy+by2+2gx+2fy+c=0 which is general equation of second degree.

Section of right circular cone by different planes

A right circular cone is as shown in the figure -1

i. Section of a right cone by a plane passing through its vertex is a pair of straight lines passing through the vertex as shown in the figure -2 .

Figure -2

ii. Section of a right circular cone by a plane parallel to its base is a circle as shown in the figure -3 .

Figure- 3

iii. Section of a right circular cone by a plane parallel to a generator of the cone is parabola as shown in the figure -4 .

Figure-4

iv. Section of a right circular cone by a plane neither parallel to any generator of the cone nor perpendicular or parallel to the axis of the cone is an ellipse as shown in the figure -5 .

v. Section of a right circular cone by a plane parallel to the axis of the cone is a hyperbola as shown in the figure -6 .

3D View :

Distinguishing various conics:

The nature of the conic section depends upon the position of the focus with respect to the directrix and also upon the value of the eccentricity.

1. If focus lies on the Directrix.

Δabc+2fghaf2b2ch2=0

General equation of a conic represents a pair of straight lines, if

  • h2>ab, the lines will be real and distinct
  • h2=ab, the lines will be coincident
  • h2<ab, the lines will be imaginary (a point)

2. If focus does not lie on Directrix.

i. Δ0,a=b,h=0, a circle

ii. Δ0,e=1, h2=ab, a parabola

iii. Δ0,0<e<1, h2<ab, an ellipse

iv. Δ0,e>1, h2>ab, a hyperbola

v. Δ0,e>1, h2>ab,a+b=0, rectangualr hyperbola

y2=4ax

Vertex (0,0)

Tangent of lotus rectum x=0

Extremities of lotus rectum (a, 2a), (a, 2a)

Length of lotus rectum. 4a

Focal distance (SP) SP=PM=x+a

Parametric form x=at2,y=2at,t is parameter.

Focal distance - the distance of a point on the parabola from the focus

Focal chord - A chord of the parabola, which passes through the focus.

Double ordinate - A chord of the parabola perpendicular to the axis of the parabola.

Latus Rectum-A double ordinate passing through the focus or a focal chord perpendicular to the axis of parabola.

  • Perpendicular distance form focus on directrix = half the latus rectum.
  • Vertex is middle point of the focus and the point of intersection of directrix and axis.
  • Two parabolas are said to be equal if they have the same lotus rectum.

Other Standard Forms of Parabola:

Equation of curve: y2=4ax x2=4ay x2=4ay
Vertex (0,0) (0,0) (0,0)
Focus (a,0) (0,a) (0,a)
Directrix xa=0 y+a=0 ya=0
Equation of axis y=0 x=0 x=0
Tangent of vertex x=0 y=0 y=0
Parametric form (at2,2at) (2at,at2) (2at,at2)

Position of a point with respet ot Parabola

S1=y124ax1 S1=y124ax1 S1=x124ay1 S1=x12+4ay1
S1<0 Inside  S1<0 Inside   S1<0 Inside  S1<0 inside 
S1>0 Outside  S1>0 Outside  S1>0 Outside  S1>0 Outside 
S1=0 on the parabola  S1=0 on the parabola  S1=0 on the parabola  S1=0 on the parabola 

Equation of Parabola when vertex is shifted.

I. Axis is Parallel to x-axis:

Let vertex A be (p,q) then equation of parabola be (yq)2=4a(xp).

II. Axis is parallel to y-axis:

Let vertex A be (p,q) then equation of parabola is (xp)2=4a(yq)

Parametric Form:

y2=4ax y2=4ax x2=4ay x2=4ay
x=at2 x=at2 x=2at x=2at
y=2at y=2at y=at2 y=at2
Parametric coordinates (at2,2at) (at2,2at) (2at,at2) (2at,at2)

Properties of Focal chord:

1. If the chord joining P(t1) and Q(t2) is the focal chord then t1t2=1.

2. Length of focal chord is a(t+1t)2

3. The length of the focal chord which makes an angle θ with the positive direction of x-axis is 4acosec2θ.

4. Semi latus rectum of a parabola is the harmonic mean between the segments of any focal chord of the parabola.

5. Circle described on the focal length as diameter touches the tangent at vertex.

Equation of tangent:

Parabola Point form Pt. of contact Parametric form Pt. of contact slope form Pt. of contact
y2=4ax yy1=2a(x+x1) (x1,y1) ty=x+at2 (at2,2at) y=mx+am am2,2am)
y2=4ax yy1=2a(x+x1) (x1,y1) ty=x+at2 (at2,2at) y=mxam (am2,2am)
x2=4ay xx1=2a(y+y1) (x1,y1) tx=y+at2 (2at,at2) x=my+am (2am,am2)
x2=4ay xx1=2a(y+y1) (x1,y1) tx=y+at2 (2at,at2) x=myam (2am,am2)
Pair of Tangents from point (x1,y1)

Let eq qn of parabola be y2=4ax

Sy24ax

S1y124ax

Tyy12a(x+x1)

Equation of pair of tangents is SS1=T2 i.e.

(y24ax)(y124a1)={y12a(x+x1)}2

Properties of Tangents:

1. Point of intersection of two tangents of the parabola:-

Equation of tangent at P is t1y=x+at12

Equation of tangent at Q is t2y=x+at22

Solving these equations, we get

x=at1t2,y=at1+at2

A(at1t2,a(t1+t2))

2. Locus of foot of prependicular from focus upon any tangent is tangent at vertex:-

Equation of tengent at P is ty=x+at2

Let the tangent meet y-axis at Q then Q(0, at )

 slope of QS=ata=t slope of tangent =1t1tx(t)=1SQ tangent 

3. Length of tangent between the pt. of contact and the point where tangent meets the directrix subtends right angle at focus:-

Equation of tangent at P(t)

ty=x+at2

Point of intersection with directrix x=a is

(a,atat)

slope SP=2atat2a=2tt21

slope QS=atat2a=t212t

m1m2=1

PSQS.

4. Tangent at extremities of focal chord are perpendicular adn intersect on directrix

(Locus of intersection point of tangents at extremities of focal chord is directrix)

Let P(at2,2at) and Q(at2,2at)

Equation of tangent at Pty=x+at2(1)

Equation of tangent at Q1ty=x+at2

y=txat(2)

Point of intersection of both tangents, we get after sloving (1) & (2) i.e.

x+a=0

A point lies on the directrix.

Example: 1 The focal chord to y2=16x is tangent to (x6)2+y2=2, then the possible value of the slope of this chord, are

(a) {1,1}

(b) {2,2}

(c) {2,12}

(d) {2,12}

Show Answer

Solution: The focus of parabola is (4,0). Let slope of focal chord be m. Equaiton of focal chord is y=m(x4). It is tangent to the circle then

|6m4mm2+1|=24m2=2(m2+1)2m2=2

m=±1

Answer: a

Example: 2 The curve represented by ax+by=1, where a,b>0 is

(a) a circle

(b) a parabola

(c) an ellipse

(d) a hyperbola

ax=1by

Show Answer

Solution: ax=1+by2by

(axby1)2=(2by)2

a2x2+b2y2+12abxy2ax+2by=4by

a2x22abxy+b2y22ax2by+1=0

Δ=|a2abaabb2bab1|=a2(b2b2)+ab(abab)a(ab2+ab2)

=02a2b22a2b2

=4a2b20

h2ab=(ab)2(a2)(b2)=0

It is a parabola

Answer: b

Example: 3 The equation of the directrix of the parabola y2+4x+4y+2=0 is

(a) x=1

(b) x=1

(c) x=32

(d) x=32

Show Answer

Solution: y2+4y=4x2

(y+2)2=4(x12)

y2=4AX

Equation of directrix is X= Ai.e. x12=1

2x3=0 or x=32

Answer: d

Example: 4 The locus of the midpoint of the segment joining the focus to a moving point on the parabola y2=4ax is another parabola with directrix

(a) y=0

(b) x=a

(c) x=0

(d) none of these

Show Answer

Solution: Let P(at2,2at) lies on the parabola

y2=4ax

Mid point of PS is Q.

h=at2+a2,k=0+2at2

2 haa=t2,ka=t

2 haa=(ka)2

a(2 ha)=k2

Locus of (h,k) is y2=2a(xa2)

Equaiton of directrix is xa2=a2

x=0

Answer: c

Example: 5 The angle between the tangents drawn from the point (1,4) to the parabola y2=4x is

(a) π6

(b) π4

(c) π3

(d) π2

Show Answer

Solution: Equation of tangent of the parabola y2=4x is

y=mx+1 m

This equation passes through (1,4) i.e.

4=m+1 m

m24 m+1=0

m1m2=1 and m1+m2=4

Angle between the two tangents is tanθ=|m1m21+m1 m2|

tanθ=|(4)241+1|=122=3

θ=π3

Answer: c

Exercise: 6 Tangent to the curve y=x2+6 at a point (1,7) touches the circle x2+y2+16x+12y+c =0 at a point Q then coordinates of Q are

(a) (6,11)

(b) (9,13)

(c) (10,15)

(d) (6,7)

Show Answer

Solution: Equation of tangent at (1,7) to the curve y=x2+6 is

y+72=x+6

2xy+5=0(1)

This line also touches the circle i.e.

Equation of normal of circle passing through

centre (8,6).

x+2y+λ=0

812+λ=0

λ=20

x+2y+20=0(2)

Q is intersection point of (1) and (2)

x=6,y=7

Q(6,7)

Answer: d

Exercise: 7 Consider the two curves c1:y2=4x,c2:x2+y26x+1=0. Then

(a) c1 and c2 touch each other only at one point.

(b) c1 and c2 touch each other only at two points.

(c) c1 and c2 intersect (but do not touch) at exactly two points.

(d) c1 and c2 neither intersect nor touch each other.

Show Answer

Solution: Let eq n of tangent of parabola be

y=mx+1 m is also a tangent to the circle then

|3m+1m1+m2|=22

(3m2+1)2m2=8(1+m2)

m42 m+1=0

m2=1m=±1 Two common tangents are possible.

Exercise: 8 If b, c are intercepts of a focal chord of the parabola y2=4ax then c is equal to

(a) bba

(b) aba

(c) abab

(d) abba

Show Answer

Solution: We know that 2a=2xSAxSBSA+SB

2a=2bcb+c

ab+ac=bc

abab=c

Answer: d

Exercise: 9 The circle x2+y22x6y+2=0 intersects the parabola y2=8x orthogonally at the point P. The equation of the tangent to the parabola at P can be

(a) 2xy+1=0

(b) 2x+y2=0

(c) x+y4=0

(d) xy4=0

Show Answer

Solution: Let y=mx+2m be tangent to y2=8x. Since circle intersects the parabola orthogonally. So this tangent is the normal for the circle. Every normal of the circle passes through its centre. So centre (1,3).

3=m+2 mm23 m+2=0

(m2)(m1)=0

m=1,2.

y=x+2 or y=2x+1

Answer: a

Exercise: 10 A tangent PT is drawn at the point P(16,16) to the parabola y2=16x. PT tangent intersect the x-axis at T. If S be the focus of the parabola, then TPS is equal to

(a) tan112

(b) π4

(c) 12tan112

(d) tan134

Show Answer

Solution: ST=4+AT=16+4=20

PS=4+16=20

TPS is isosceles triangle

tan2θ=160164=43=2tanθ1tan2θ

2tan2θ+3tanθ2=0

(2tanθ1)(tanθ+2)=0

tanθ=12,tanθ=2 (Not possible)

θ=tan112(θ is acute angle )

Answer: a

Exercise: 11 If a,b,c are distinct positive real numbers such that the parabolas y2=4ax and y2 =4c(xb) will have a common normal, then

(a) 0<bac<1

(b) bac<0

(c) 1<bac<2

(d) bac>2

Show Answer

Solution: Equation of normals are

y=mx2amam3.(1)y=m(xb)2cmcm3.(2)

Equation 1 and 2 are identical then

2amam3=bm2 cmcm3÷m

2a+am2=b+2c+cm2

(ac)m2=b+2(ca)

m2=bac2

m=±bac2

For m be real bac>2

Answer: d

Exercise: 12 If AB be a chord of the parabola y2=4ax with vertex at A.BC is perpendicular to AB such that it meets the axis at C. The projection of the BC on the axis of parabola is

(a) 2a

(b) 4a

(c) 8a

(d) 16a

Show Answer

Solution: Let coordinates of B be (x,y)

In ABD,tanθ=BDAD=yx

In BCD,tan(90θ)=BDDC

DC=yyx=4axx=4a

Answer: b

Exercise: 13 A circle is descirbed whose centre is the vertex and whose diameter is three-quarters of the latus rectum of the parabola y2=4ax. If PQ is the common-chord of the circle and the parabola and L1 L2 is the latus rectum, then the area of the trapezium PL1 L2Q is

(a) (2+22)a2

(b) 4a2

(c) 22a2

(d) 32a2

Show Answer

Solution: Centre of circle (0,0)

diameter =344a=3a

Eq of circle x2+y2=9a24.(1)

Eqn of parabola y2=4ax.(2)

coordinates of P and Q, we get after solving (1) and (2).

x2+4ax=9a24

(x+2a)2=(5a2)2x=2a±5a2

x=a2,9a2( not possible )

P(a2,2a),Q(a2,2a)y=±2a

PQ=22a,L1 L2=4a

Area of trapezium =12(PQ+L1 L2)x distance between them.

=12(22a+4a)×(aa2)=(2+2)2a2

Answer: a

Exercise: 14 From the point (15,12) three normals ae drawn to the parabola y2=4x, then centroid of triangle formed by three co-normal points is

(a) (5,0)

(b) (5,4)

(c) (9,0)

(d) (263,0)

Show Answer

Solution: Let equation of normal be y=tx+2t+t3

It passes through (15,12). So 12=15t+2t+t3

t313t12=0(t+1)(t+3)(t4)=0t=1,3,4

Points are (at2,2 at) i.e. (1,2),(9,6),(16,8)

Centroid is (1+9+163,26+83)=(263,0)

Answer: d

Exercise: 15 A ray of light travels along a line y=4 and strikes the surface of a curve y2=4(x+y) then equation of the line along reflected ray travel is

(a) x+1=0

(b) y2=0

(c) x=0

(d) x2=0

Show Answer

Solution: y24y=4x

(y2)2=4(x+1)

Focus (0,2)

Incident ray is parallel to axis of the parabola, so reflected ray passes through focus (0,2) i.e. x=0

Exercise: 16 Let P be a point of the parabola y2=3(2x3) and M is the foot of perpendicular drawn from P on the directrix of the parabola, then length of each side of an equilateral triangle SMP, where S is focus of the parabola is

(a) 6

(b) 8

(c) 10

(d) 11

Show Answer

Solution: Equation of parabola is

y2=6(x32)

focus S(3,0)

equation of directrix x=0

P(32+32t2,3t)

Coordinates of M(0,3t)

MS=9+9t2

MP=32+32t2, But MS=MP

9+9t2=94+94t2+94t2

t2=3

Length of side =6

Answer: a

Exercise

1. The point (2a,a) lies inside the region bounded by the parabola x2=4y and its latus rectum. Then

(a) 0<a1

(b) 0<a<1

(c) 0a1

(d) a<1

Show Answer Answer: b

2. Two perpendicular tangents to y2=4ax always intersect on the line

(a) x+a=0

(b) xa=0

(c) x+2a=0

(d) y+2a=0

Show Answer Answer: a

3. C1:y2=8x and C2:x2+y2=2. Then

(a) C1 and C2 have only two common tangents which are mutually perpendicular

(b) C1 and C2 have two common tangents which are parallel to each other.

(c) does not have any common tangent.

(d) C1 and C2 have four common tangents.

Show Answer Answer: a

4. Two common tangents to the circle x2+y2=2a2 and y2=8ax are

(a) y=±(x+a)

(b) x=±(y+2a)

(c) y=±(x+2a)

(d) x=±(y+a)

Show Answer Answer: c

5. The number of points with integral coordinates that lie in the interior of the circle x2+y2=16 and the parabola y2=4x are

(a) 6

(b) 8

(c) 10

(d) 12

Show Answer Answer: b

6. The vertex of the parabola x2+y22xy4x+4=0 is at

(a) (12,12)

(b) (1,1)

(c) (1,1)

(d) (+12,12)

Show Answer Answer: d

7. The length of the latus rectum of the parabola Missing or unrecognized delimiter for \left is

(a) 2a

(b) 2a

(c) 22a

(d) 32a

Show Answer Answer: c

8. The point on y2=4ax nearest to the focus is

(a) (0,0)

(b) (a,2a)

(c) (a,2a)

(d) (a4,a)

Show Answer Answer: a

9. The angle between the tangents drawn from the origin to the parabola y2=4a(xa) is

(a) 45

(b) 60

(c) 90

(d) tan112

Show Answer Answer: c

10. The circle x2+y2+2λx=0,λεR, touches the parabola y2=4x externally. Then

(a) λ>0

(b) λ<0

(c) λ>1

(d) none of these

Show Answer Answer: a