ANALYTICAL GEOMETRY Definition of Conic: The curves obtined by slicing the cone with plne not pssing through the vertex re clled conics. A Conic is the locus directrix of point which moves in plne, so tht its distnce from fixed point bers constnt rtio to its distnce from fixed stright line. The fixed point is clled focus nd the fixed stright line is clled directrix nd the constnt rtio is clled eccentricity which is denoted by e FP PM = Constnt = e Generl eqution conic: Ax + Bxy + Cy + Dx + Ey + F = 0 1) A prbol if B - 4AC = 0 ) An ellipse if B - 4AC < 0 3) A hyperbol if B - 4AC > 0
Clssifiction of conics with respect to eccentricity: 1) If e < 1, then the conic is n ellipse ) If e = 1, then the conic is prbol 3) If e > 1, then the conic is hyperbol Open rightwrd: y = 4x STANDARD PARABOLAS Focus: The fixed point used to drw the prbol is clled the focus F focus is F(,o) Directrix: The fixed line used to drw prbol is clled the directrix of the prbol. Eqution of the directrix is x = Axis: The xis of the prbol is the xis of symmetry. The curve y =4x is symmetricl bout x xis nd hence x xis or y=0 is the xis of the prbol.
Axis of the prbol psses through the focus nd is perpendiculr to the directrix. Vertex: The point of intersection of the prbol nd its xis is clled its vertex. Vertex is V(0,0) Focl Distnce: Focl distnce is the distnce between point on the prbol nd its focus. Focl Chord: A chord which psses through the focus of the prbol is clled the focl chord of the prbol. Ltus Rectum: It is focl chord perpendiculr to the xis of the prbol. Here the eqution of the ltus rectum is x = End point of Ltus Rectum y =4x x = y = 4. = 4 y = ±
L is (, ) is (, -) The length of Ltus rectum is LL =4
OTHER STANDARD PARABOLAS 1. Open Leftwrd: y = - 4x > 0 Focus F (-, 0) Vertex V (0, 0) Axis x xis, y =0 Directrix x= Eqution of Ltus rectum x = - Length of Ltus rectum = 4
. Open Upwrd: x = 4y > 0 Focus F (0, ) Vertex V (0, 0) Axis y xis, x =0 Directrix y = - Eqution of Ltus rectum y = Length of Ltus rectum = 4
3. Open Downwrd: x = -4y Focus F (0, -) Vertex V (0,0) Axis y xis x =0 Directrix y = Eqution of Ltus rectum y = - Length of Ltus rectum = 4 If the vertex of the prbol is V(h, k) then Eqution of the prbol (y - k) = 4(x - h) (open rightwrd) (y - k) = -4(x - h) (open leftwrd) (x - h) = 4(y - k) (open upwrd) (x - h) = -4(y - k) (open downwrd)
ELLIPSE: Stndrd eqution of the ellipse x +y b = 1 > b Focus: The fixed point is clled focus denoted by F1(e, 0).By symmetry we hve F(-e, 0). Directrix: The fixed line is directrix eqution is x =.By symmetry nother e directrix is x= - e Mjor xis: The line segment AA is mjor xis nd length of mjor. Eqution of mjor xis is y = 0. xis is Minor xis: The line segment BB is clled the minor xis nd length of minor xis is b. Eqution of minor xis is x = 0. Length of mjor xis is lwys greter thn minor xis.
Centre: The point of intersection of mjor xis nd minor xis is clled the centre of the ellipse. C(0, 0) is centre. Centre need not to be origin. End points of Ltus rectum: L1 is e, b, L1 is e, b The end points of the other Ltus rectum re (-e, ± b ) Length of Ltus rectum is b Vertices: The points of intersection of the ellipse nd mjor xis re clled the vertices. A(, 0), A (-, 0) re vertices. Focl distnce: The focl distnce with respect to ny point p on the ellipse is its distnce of p from the referred focus. Focl chord: A chord which psses through the focus of the ellipse is clled the focl chord of the ellipse. Ltus rectum: It is the focl chord perpendiculr to the mjor xis of the ellipse. The eqution of ltus rectum re x= e, x=-e Eccentricity e = b
The other stndrd form of the ellipse is x +y = 1 b >b The mjor xis of the ellipse is long y xis Centre C (0, 0) Vertices A (0, ) & A (0, -) Foci F1(0, e) & F(0, -e) Eqution of mjor xis x = 0 Eqution of minor xis y = 0 End points of minor xis B(b, 0) & B (-b, 0) End of directrices y = ± l End points of Ltus rectum: ± b, e ± b, e
Generl form of stndrd ellipse If centre is C(h, k) The generl form of stndrd ellipses re: (x h) + (y k) b = 1 nd (x h) b + (y k) = 1 >b Focl property of n ellipse: The sum of the focl distnces of ny point on n ellipse is constnt nd is equl to the length of mjor xis. To Prove: F1p + Fp = Let p be ny point on the ellipse.drw perpendiculr PM nd PM to directrices x= e nd x= - e
F 1 P PM = e, F P PM = e F 1 p = epm, F p = epm F 1 p +F p = e (PM+PM ) = e MM = e. e = = length of mjor xis Hyperbol: x y - b = 1
Focus: The fixed point is clled focus F1(e, 0) nother focus is F(-e, 0). Directrix: The fixed line is directrix eqution is x = nother directrix is e x = - e Trnsverse xis: The line segment AA joining the vertices is clled the trnsverse xis nd the length of the trnsverse xis is. The eqution of the trnsverse xis is y = 0.Note tht the brnches of the curve. Conjugte xis: The line segment joining the points B(0,b) nd B (0,-b) is clled conjugte xis. The length of conjugte xis is b. The eqution of the conjugte xis is x = 0. Centre: The point of interction of trnsverse xis nd conjugte xis of the hyperbol is clled centre of the hyperbol C(0, 0) is centre. Vertices: The point of intersection of the hyperbol nd its trnsverse xis is clled the vertices. The vertices of hyperbol A(, 0) nd A (-,0). Eccentricity : e = +b Ltus rectum: It is focl chord perpendiculr to the trnsverse xis of the hyperbol. The eqution of the ltus rectum re x = ± e. End points of Ltus rectum: e, b, e, b (-e, ± b ) Length of Ltus rectum is b
The other form of hyperbol y x - b = 1 Centre C(0, 0) Vertices A(0,) & A (0,-) Foci F1(0,e) & F(0,-e) Eqution of trnsverse xis x = 0 Eqution of conjugte xis y = 0 End points of conjugte xis (b, 0) & (-b, 0) Eqution of Ltus rectum y = ±e Eqution of Directrices y = ± e
End points of Ltus rectum: ± b, e ± b, e If the centre is (h, k) then stndrd equtions of hyperbol re (x h) (y k) b = 1 nd (y k) (x h) b = 1 Prmetric form of conics: Conic Prmetric equtions Prbol x = t y = t Ellipse x = cos θ y = b sin θ Hyperbol x = sec θ y = b tn θ Prmeter Rnge Any point on conic t - < t < t or (t, t) θ 0 θ θ or (cos θ, bsinθ) θ 0 θ π θ or (sec θ, btn θ) Note: For ellipse nother prmetric equtions x = 1 t, y = b t 1+t 1+t - < t < Note: The eqution of tngent t (x 1,y 1 ) is obtined from the eqution of the curve by replcing x by xx 1, y by yy 1, xy by 1 xy 1 + yx 1 x by 1 x + x 1 nd y by 1 y + y 1
Condition for y = mx + c to be tngent to the conics: 1. Prbol: (i) The condition for y=mx+c to be tngent to the prbol (ii) y = 4x is c = m The point of contct is m, m (iii) The eqution of ny tngent is y = mx + m. Ellipse: (i) The condition tht y= mx + c my be tngent to the ellipse x + y b = 1 is c = m + b. (ii) The point of contct is m c, b c (iii) The eqution of ny tngent is of the form y = mx ± m + b 3. Hyperbol : (i) The condition tht y = mx+c my be tngent to the hyperbol is c = m - b. (ii) The point of contct is m c, b c (iii) The eqution of ny tngent is of the form y = mx ± m b Results relted to conics without proof: 1. The tngents t the ends of the focl chord intersect t right ngles on the directrix of the prbol.. If the tngent nd norml t point P on the prbol y=4x meets the xis of the prbol t T nd respectively nd F is the focus then FT = FA = FP
3. If FY nd F Y re perpendiculrs from the foci F nd F of n ellipse on the tngent t ny point then FY.F Y = b. 4. If the norml t n end of ltus rectum of the ellipse psses through n end of the minor xis then e 4 + e - 1= 0. 5. If the tngent t ny point P on the ellipse x + y b = 1 whose centre is C meets mjor xis t T nd PN is the perpendiculr to the mjor xis then CN.CT =. 6. If the norml t the end of ltus rectum of the ellipse x + y b = 1 intersects the mjor xis t then CG=e 3 where C is centre of ellipse. 7. If F 1 nd F re the foci nd B nd B re the end points of the minor xis of n ellipse then F 1 B F B is rhombus nd its re is be. 8. If the norml t the end of the ltus rectum of the hyperbol x y b =1 intersects the trnsverse xis t then CG=e 3, C is centre. 9. If the tngent t ny point P of the hyperbol x y b = 1 whose centre is C meets the trnsverse xis t T nd PN is the perpendiculr to the trnsverse xis then CN.CT =.
Asymptotes: An symptote to curve is the tngent to the curve such tht the point of contct is t infinity. The symptote touches the curve t + nd -. The eqution of symptotes to the hyperbol x y b =1 Assume the eqution of symptote is of the form y = mx + c x y b =1 is hyperbol Solving, x mx + c b = 1 1 m b x mc b x c b + 1 =0
The points of contct re t infinity ie. The roots of the equtions re infinite. Since the roots re infinite the coefficients of x nd x must be zero. The combined eqution is 1 m b = 0, mc b = 0 m = ± b c = 0 x + y b = 0, x y b = 0 y = ± b h The combined eqution is x y b =1 Results: 1. The symptotes pss through the centre C(0,0) of the hyperbol.. The slopes of symptotes re b nd - b ie., the trnsverse xis nd conjugte xis bisect ngles between the symptotes. 3. If α is the ngle between the symptotes tn α = b 4. Sec α = 1+ tn α α = tn -1 b = 1+ b = +b = e Sec α = e α = sec 1 e Angle between symptotes α = sec -1 e
5. The stndrd eqution of hyperbol nd combined eqution of symptotes differ only by constnt. Rectngulr Hyperbol: A hyperbol is sid to be rectngulr hyperbol if its symptotes re t right ngles α = tn 1 b = 90o tn 1 b = 90o tn 45 = b = b x y =1 is b x y = The combined eqution of symptotes is x y = 0 The symptote eqution re x y = 0, x + y = 0. The trnsverse xis is x = 0 nd conjugte xis is y = 0. If the symptotes re co-ordinte xes, then the rectngulr hyperbol is stndrd one.
For stndrd rectngulr hyperbol the symptotes re coordintor xes. Eqution of the symptotes re x = 0,y = 0. The eqution of stndrd rectngulr hyperbol is xy = k. Let the symptotes meet t c. Let AA = be the length of trnsverse xis. Drw AM perpendiculr to x xis. ALM = 45 o CM = cos 45 o = AM = sin 45 o = A is, xy = k k =. xy = = let c = xy = c is the eqution of stndrd rectngulr hyperbol. 1. Eccentricity of rectngulr hyperbol is. Vertex re,,, 3. Foci re (, -), (-, -) 4. Eqution of trnsverse xis is y = x nd conjugte xis is y = -x 5. If (h, k) is centre, eqution of the curve is (x - h)(y - k) = c 6. Prmetric eqution of rectngulr hyperbol xy = c re x = ct nd y = c t. Any point of the rectngulr hyperbol is ct, c t
Results: 1. Eqution of tngent t (x 1, y 1 ) to the rectngulr hyperbol xy = c is xy1 + yx1 = c. Eqution of tngent t t is x + yt = ct 3. Eqution of norml t (x 1, y 1 ) is xx1 - yy1 = x1 y1 4. Eqution of norml t t is y - xt = c t ct3 5. Two tngents nd four normls cn be drwn from point to rectngulr hyperbol.