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Conic sections
Mathematics class 11 (041)
University of Kerala
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CONIC SECTION
In NDA exam, generally 2-4 questions are asked from this chapter which are based on finding equation of conics, eccentricity conics, various terms related to conics etc.
A conic section or conic is the locus of a point which moves in such a way that its distance from a fixed point always bears a constant ratio to its perpendicular distance from a fixed line, all being in the same plane. Shape of the conic section obtained depends on the position of cutting plane. This section deals with parabola, hyperbola and ellipse etc.
Various Terms Related to Conic Section 1. Focus The fixed points is called the focus of the conic section. 2. Directrix The fixed straight line is called the directrix of the conic section. 3. Eccentricity The constant ratio of distance of point lying on conic,from focusto its perpendicular distance from directrix is called the eccentricity of the conic section and is denoted bye. (i)Foran ellipse, e < 1 (iv) For a circle, e = 4. Axis The straight line passing through the focus and perpendicular to the directrix, is called the axis of the conic section.
(i) For a parabola, e = 1 () Pair of straight lines, e = oo
(ii) For hyperbola, e >
- Vertex The point of intersection of the conic section and its axis, is called vertex of the conic section.
- Centre The point which bisects every possible chord of the conic passing through it, is called the centre of conic.
- Latusrectum The latusrectum of a conic is the chord passing through the focus and perpendicular to the axis. . Focal chord Any chord passing through the focus of a conic is called the focal ehord of the conic. . Double ordinate Any chord perpendicular to the axis of a conic is called the double ordinate of that conic. . Focal distance or focal length The distance between the focus and a point lying on the conic, is known as focal distance or focal length of the given point.
Eneral Equation of Conics of Second Degree veneral equation of conics of second degree viz.
ax2hxy + by' +28x + 2/y+c= here, discriminant A abc+ 2/gh- af- bg -ch
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- Form x = 4ay In this form the focus of the
parabola lie in the positive
The above equation represents a non-degenerate side of Y-axis conic whose nature is as follows
S. Condition Nature of conic A0,h= 0, a- b, e =0 A circle S (0,a) A0, ab -h* = 0, e= 1 A parabola A0, ab -h> 0, e<1 An ellipse X A0, ab-h<0,e>1 A hyperbola o 0) A0, ab -h* < 0, a + b = 0 = 1/
A rectangular hyperbola y=-aa V x= 4ay PARABOLA A parabola is the locus of a point which is equidistant from a fixed point called focus and from a fixed straight line called directrix.
- Form x = - 4ay In this form the focus of the
parabola lie in the negative of Y-axis.
Directrix x. y) L(a, 2a) O (0, 0) A M Latusrectlum X Verlex Focus axis X 0) Sa, 0) S (0. a) y = 4aax Aay ' (a,-2a) Some Important Results Related to Parabola 2 -4 ay
Since, point P(x, y) lies on the parabola. PS = e = 1 PM
here, e = 1 y= 4ax y? = - 4 ax 2=4 ay
PS = PM12 Eccentricity e=1 e e 1 e = Coordinates of (0, 0) (x-a) +(y-0) = (x + a) = 4Aaxx
,0) (0, 0) ,0) vertex Coordinates of focus , 0) (-a, 0) (0, a)
(0, a) which is the equation of parabola in standard form. Equation of the directrix
y= -a y = a Noto The distance between vertex of parabola and focus is equal to perpendicular distance of vertex from directrix. or Vertex is the mid-point of line joining focus and point of intersection of directrix and axis.
Equation of the axis y = 0 y=0 x = 0 x = 0
Length of the latusrectum 4a 4a 4a 4a Other Standard Forms of Parabola
- Form y' = - 4ax In this form the focus of the parabola lie in the negative side of X-axis.
Focal distance of a point a-y Plx)
a-: y+ a
Directrix Extremities of (a, t 2a) (-a,t 2a) (t 2a,a) (t2a,-a) latusrectum Parametric (at", 2at) (-at ?, 2at) (2at, at ) (2a-a X S a, 0) coordinates o0,0) X a Parametric x = at x = -at t = 2at, *=2al equation y= 2at y= 2at y at y= -at y'- 4ax
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So, the PQ and AP are perperndicular to each other
-1 Y-2= -} t2 1 y4.
Sol. c. Let S = y* +9- 6y5x Now, point (1, 4) lies on S. Thus, S = (4) +9 64) -5(1) = 16+9-24 - 5 From Eqs. (i) and (ii), Y 4 and 2
So, the coordinates of point ( are (8,4).
25 29=- 4 < Hence, the point lies inside the parabola.
Required distance, PQ V(
4-4- =364 36= 72= 6/ Equation of Chord The equation of chord joining the points (x, y1) and (x22) on the parabola y* = 4ax is
y(yi +y2)= 4ax+ y12 ELLIPSE
An ellipse is the locus of that pont which moves in a
Condition for the Chord to be a Focal Chord plane such that the ratio of its
distance from a fixed
The chord joining the points (ati, 2at,) and (at, 2at) passes through focus provided tfa =-
point (focus) to a fixed straight
line (directrix) is constant and less than unity. This constant ratio is called
eccentricity and denoted by e. Or
####### Tangents and Normals to the Parabola The locus of a
point whose sum of distance from two Equation of tangent to the parabola y* = 4ax fixed points remains constant is called an cllipse. ) at (x. y1) is y yi =2a(x +x) (11) at (at , 2at ) is ty = x+ at
(ii) If m is the slope of tangent to the parabolay = 4ax, then
Minor ax1S B (0,D p y) V
a, O. its equationis y= nmx+. Point of contact is X+- S'-ae, 0) O(0, 0) S (ae, 0)/Majo m axis X Equation of normal to the parabola » = 4ax |Directrices B (0, -b)
Drectrices (4) at (x1, 1 ) is y- i =-x-x) 2a PS = e, PS = e'PM, (r - ae) + (y-0) = e i) at (at', 2at ) is y+ tx PM = 2at+at
(11) If m is the slope of normal to the parabola y= 4ax, then its equation is given as y = mx - 2am -m°. d b Where, b' =a* (1- e), b < a EXAMPLE 4. Let P(2, 2) is a point on the parabola y' =2x and A is its vertex. If Q is another point on the
This is the equation of ellipse in standard form. Here AA = 2a and BB = 2b are called major and minor axes
parabola such that PQ is perpendicular to AP, theen respectively. what is the length of PQ? Another Form of Ellipse
The another equation of the ellipse is where, b> a).
a. 2 b. 2/2 c. 42 d. 62 Sol. d. Equation of parabola is y = 2x A y=b/e Directrix Major axis
X A(0. 0) S X be A O(O. 0) a. O) 0-be Minot axis
y Q Y) a,0)
X So, the coordinates of vertex are Al0, 0).
Let (,Y) be the coordinates of the point Q yi 2x . .(0) B -b/e and slope of PQ = 2 2
y Directrix
Also, slope of AP =- 2-
Here, AA' = 2a and BB' = 2b are minor and major
axes, respectively.
####### MATHEMATICS> Conic Section 209
Came important Results Related to Ellipse (ii) outside the ellipse, if S,> 0
b ,ab (ii) inside the ellipse, if S, <0 where, 5,
####### Equation of Ellipse in
Parametric Form
Cordinatos of the (0, 0) (0, 0) Cordinate% of the (a, 0) and (-a, 0) (0, b) and (0, b) verticss Coordinates of foci (a6, 0) and (-a0, 0) (0, be) and (0, be) Parametric equation of ellipse+71 is Lonigth of rrajor axis 2a 2b x = a cos G, y = bsin0 and parametric coordinates of point lying on it is given by (a cos 0, b sin 6). The angle 6 is called the eccentric angle of the point (a cos 0, bsin 6) on the ellipse.
Lrngth of minor axis 2b 2a Equalion of the major
Equation of the rminor y Equation of Chord Equaion of the and x and y m - drectrices QU2 Y2)
Eccentricity a P1, Y1) Length of latusrecturn 2b 2a a b Foral distances of a R3 Y3) t (z. Y) at 6x biey The equation of the chord of the ellipse Enremities of y alurecturn t 1, whose mid-point be (x1, y1) is T = S1, a' Hote If th6 Centre of ellipse is at (h, k) and its axis are parallel to the coordinate axes, then the equation of ellipse is given as x-hf(y-k 1
here T- -1,5,--
The equation of the chord joining the points P(acos 0, bsin 6, ) and Q(a cos 02, bsin 0,) is
b EXAMPLE 5. The equation of the ellipse, whose centre is
a origin (0, 0), foci (t1,0) and eccentricity,
is
Tangents and Normals to the Ellipse
####### 12
d. x'+y-12 Equation of tangent to the ellipse
O1 b. Here, foci = (+1,0) and eccentricity, e= (i) at the point (x, % ) is 1 ae=1 and e= a= (n) having slope m is y = mxtva' m +b and coordinates of point of contacts are a m
b'-a (1-e)b-41J-4x
So, the equation of required ellipse is Va m+b
Position of a Point with
Kespect to an Ellipse
Equation of normal to the ellipse
####### HS -1 =0. is the equation of an ellipse, then a i) at the point (x1, ) is-2=a' - b?.
b nt
####### Pz ) in the XY-plane lies
) on the ellipse, if S, = 0
(1) having slope m is y= mx+ m(at - b*) Na+b'm
211 er MATHEMATICS Conic Section
hyperbola is at point (h,k) and ta axe are parale to the coordinate axes, then the equation of (y- D
Let 2QCN = e Here, P and Q are the corresponding points on the hyperbola and the auxiliary circle (0 s6 2)
Note centre of
yperbola is given as h a
Equation of Hyperbola in Parametric Form
Equation of Chord The equation of chord joining the points P = (a sec 6,, ban6,) and Q= (a sec 6,, btan 8, ) is
Parametric equation of the hyperbola 1, is
asec y = btan 0 and parametric coordinates of point lying on it is given by (a sec 6, b tan 9) Tangents and Normals
Position of a Point with = Equation of tangent to the hyperbola Respect to a Hyperbola Let the equation of hyperbola be S - 1 -0 (0) at the point (. y )s
() having slope m Ls y = mx t va'm Then, the point P(x, ) lies ) on the hyperbola, if S, - 0 (i) outside the hyperbola, if S, > inside the hyperbola, if S, < 0
1 Equation of normal to the hvperbola
)at the pount (,. , )s
where, S, (n) having slope ms y = x intersection of a Line and a Hyperbola The straught line y mx+ Will cut the hyperbola EXAMPLE 7. f the eccentricity and length of 1 and 10 units I n twa points w h1ch may be real, coincident latusrectum of a
hypertbola
are
anduns
respectively, what is the length of the transverse axis
b unts
or maganary, as (>1- 1<a'm-b 15 units unts units Auxiliary Circle of Hyperbola Sol. eHere e andlength of latusre Tum
10
We knowthal
1515 he the hyperbola, then equatson of the Lergth of transverr axis
ary 1 le it
Conic sections
Course: Mathematics class 11 (041)
University: University of Kerala
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