Conic Sections- Complete Info

8/12/2019 Conic Sections- Complete Info

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Sri Chaitanya IIT Academy., A.P. CONIC SECTIONS TOTAL INFORMATION

>>1 1, then the locus of P is called a Hyperbola.

OBSERVATIONS :

1. A conic is a second degree non-homogeneous equation in x and y.

2. A second degree non homogeneous equation in x and y, Ax2 + 2Hxy + By2 + 2Gx + 2Fy + C = 0represents.

1. Pair of lines : If, = 0, H2 AB, G2 AC, F2 BC.

2. Circle : If, 0, A = B, H = 0., G2 + F2 - AC 0

3. Parabola : If, 0, H2 = AB.4. Ellipse : If 0, H2 < AB.5. Hyperbola : If 0, H2 > AB.

THE PARABOLA DEFINITIONS, FORMULAE, FACTS ON PARABOLA

1. If SP/PM = 1, then the locus of P is called a parabola.2. The standard form of parabola is y2 = 4ax.3. Different forms of Parabolas :

(i) y2 = 4ax ; a > 0 This is a parabola, whose axes is along x-axes.

(ii) y2 = 4ax ; a < 0 This is a parabola whose axes is along x-axes.

MP

l (fixed line)

S (fixed)

Y

X A

Y

X A


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>>2 0 This is a parabola whose axes is along y-axes.

(iv) x2 = 4ay ; a < 0 This is a parabola, whose axes is along y-axis

(v) (y - )2 = 4a(x - ) (or) x = ly2 + my + n. This is a parabola,

whose axes is parallel to x-axes.

(vi) (x - )2 = 4a(y - ) (or) y = lx2 + mx + n. This is a parabola,

whose axes is parallel to y-axes.

4. Parabola is not a closed curve.5. Axes of the parabola is the line where it is symetrical about it.6.

From the above diogram1. The line ZM r to x-axes (Axes of the parabola) is called directrix.

2. The line which is r to the axes and passing through the focus S is called latus rectum. From the diagram,LL is latus rectum.

3. The chord which passes through the focus of the parabola is called focal chord. From the above diagram,QR is focal chord.

4. The line passing through P and r to axes of the parabola is called double ordinate. From the abovediagram, PT is double ordinate.

Y

X A

Y

X A

Y

X

A

y

A

x

Y

X

A

y

A

x

-

-

-

-

-

-

-

-

-

-

-

-

-

-

y

y

XES

P

LQ

RL

A

T

Z

MY

l


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>>3 0, S 11 < 0, S 11 = 0 respectively.

9. Equation of the tangent to the parabola at P(x 1 , y1 ) is yy 1 - 2a(x + x 1 ) = 0.

10. Condition at which the line y = mx + c is tangent to the parabola y 2 = 4ax is c = a/m

11. Equation of any tangent to the parabola y 2 = 4ax is y = mx + (a/m)

12. If the line y = mx + c is tangent to the parabola y 2 = 4ax, the point of contact is (c/m, 2a/m) (or) (a/m 2 , 2a/m)

13. The condition at which the line lx + my + n = 0 may be a tangent to the parabola y 2 = 4ax isln = am 2

14. Point of contact of the line lx + my + n = 0 w.r.t the parabola y 2 = 4ax isn , - 2aml lHG K J

15. Equation of the normal to the parabola y 2 = 4ax is y = mx - 2am - am 3 . Where m is slope of the normal to the parabola.

16. Atmost two tangents are possible to draw to a parabola.17. Equation of pair of tangents to the parabola y 2 = 4ax at (x 1 , y1 ) is S 1

2 = S.S 11

18. Equation of the chord joining the points (x 1 , y1 ) and (x 2 , y 2 ) on the parabola y 2 = 4ax is

S1 + S 2 = S 12 (or) (y - y 1 ) (y - y 2 ) = y 2 - 4ax.

19. The locus of point of intersection of r tangents to the parabola is directrix

20. The line y = mx + c is tangent to the parabola y 2 = 4a(x + a) then the condition is c = am + (a/m)

21. Locus of the foot of the r from the focus to the tangent of the parabola y 2 = 4ax is tangent at

the vertex (i.e., y-axes).

22. [The locus of the foot of the r from focus on a tangent to along the parabola].

Equation of the chord of contact of the point (x 1 , y1 ) w.r.t. the parabola y 2 = 4ax is yy 1 - 2a(x + x 1 ) = 0.

23. Equation of the polar w.r.t the point P(x 1 , y1 ) to the parabola y 2 = 4ax is y y 1 - 2a(x + x 1 ) = 0

24. Pole of the line lx + my + n = 0 w.r.t the parabola y 2 = 4ax isn , - 2aml lHG K J

25. If the lines l 1 x + m 1 y + n 1 = 0, l 2 x + m 2 y + n 2 = 0 are conjugate w.r.t. the parabola y 2 = 4ax, is

n 2 l 1 + n 1 l 2 = 2am 1 m 2 .


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>>17 0

h > 2a

Hence abscissa of the point of concurrency of 3 concurrent normals > 2a.

Prob.1 Find the locus of a point which is such that(a) two of the normals drawn from it to the parabola are atright angles, (b) the three normals through it cut the axis in points whose distances from the vertex arein arithmetical progression.

[Ans : (a) y2 = a(h 3a) ; (b) 27ay2 = 2(x 2a)3 ]

Sol. (a) we have m1 m2 = 1

also m1 m2 m3 = ak

m3 = ak

put m3 = ak

is a root of

am3 + (2a h)m + k = 0


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>>31 0.

iii) lies on the hyperbola S=0 if S 11

= 0.

10. Two tangents can be drawn to a hyperbola from an external point.

11. The equation of the tangent to the hyperbola S=0 at P(x1 ,y

1) is S

1 = 0.

12. The equation of the normal to the hyperbolax2

a2

y2

b2 = 1at P (x

1 , y

1) is

a2xx1

b2yy1

a2 b2

13 . The condition that the line y = mx+c may be a tangent to the hyperbolax2

a2

y2

b2 = 1 is c

2 = a 2m2 b2 and the point

of contact is

c b

cma 22

, .

14 . The condition that the line lx+my+n = 0 may be a tangent to the hyperbolax2

a2

y2

b2 = 1 is

a 2l2 b 2m2=n 2 and the point of contact is

nm b

na 22 ,l .

15. The equation of a tangent to the hyperbolax2

a 2

y2

b2=1 may be taken as y = mx a2m2 b2 .

16. If m1 , m

2 are the slopes of the tangents through P to the hyperbola

x2

a 2

y2

b2 = 1 then

m1+m

2 =

2x1y1x1

2 a2 ; m 1m2 =y1

2 b2

x12 a2 .

17. If is the angle between the tangents drawn from a point (x1 , y

1) to the hyperbola S= 01

by

ax

2

2

2

2

then tan = 2221

21

112ba y x

S ab


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>>33 k > b > 0) are confocal

and therefore orthogonal.

HL 4 The foci of the hyperbola and the pointsP and Q in which any tangent meetsthe tangents at the vertices areconcyclic with PQ as diameter of the

circle.


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Sri Chaitanya IIT Academy., A.P. CONIC SECTIONS TOTAL INFORMATIONStep - 3 :

Now the equation ......(1) is in the form

2

22 2 3

2 f gY X

Length of the latus rectum is

2

2 2 3

2 f g

Remember :

i) Latus rectum of the parabola 22 2 2 2a b x y bx ay ab is 2 22ab

a b

ii) Lengths of the semi axis of the conic 2 22ax hxy ay d ared d

and a h a h respectively

and their equation is x 2 y 2 = 0

iii) The squares of the semi axis of the conic 2 22 2 2 0ax hxy by gx fy c are

22 22

4ab h a b a b h

where abc + 2fgh af 2 bg 2 ch 2

iv) Eccentricity of the conic2 2

2 2 2 0ax hxy by gx fy c is

24

22

41

a be

e ab h

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Conic Sections- Complete Info

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