GRAPHS OF CONIC SECTIONS OR SECOND DEGREE CURVES

This animation conveys that circle, ellipse, parabola and hyperbola all belong to the same family of curves, called the conics.

Conics are named as such because these are cross sections or curves of intersection obtained when we use a plane to cut or intersect a right circular cone of two nappes at different angles.

ELLIPSE PARABOLA HYPERBOLA CIRCLE

1. Parabola 2. Ellipse Circle 3. Hyperbola CONIC SECTIONS 1. Parabola 2. Ellipse Circle 3. Hyperbola either A or C is equal to 0 A & C have the same signs A = C A & C have opposite signs Second Degree Equation in Two Variables B = 0 , x-axis and y-axis are NOT rotated.

Determine the conic section that is described by the following equation. 𝑥 2 =24𝑦 𝑥 2 + 𝑦 2 =300 𝑥 2 −4 𝑦 2 +4𝑥+24𝑦+28=0 25 𝑥 2 −16 𝑦 2 =400 5 𝑥+3 2 +5 𝑦−7 2 =100 16 𝑦 2 +25 𝑥 2 =100 25 𝑥 2 +16 𝑦 2 −150𝑥−128𝑦+381=0

Latus rectum of length 4p PARABOLA is the set of all points P equidistant from a fixed point F and a fixed line D. V --- vertex Q F V D F --- focus L 2p D --- directrix axis of parabola p p 2p p = distance from V to F R Latus rectum of length 4p

Consider a parabola with vertex V = (h , k) & axis parallel to x-axis x – h D k h Q(x , y) V F (h + p, k ) x y p By definition of parabola , QF = QD

STANDARD EQUATIONS OF PARABOLA Vertex ( h , k ) axis // x-axis (one value of x corresponds to 2 values of y) V(h,k) V(h,k) V(h,k) V(h,k) axis // y-axis (one value of y corresponds to 2 values of x )

General Form of Equation of PARABOLA

1. Examples V = (2 , 4) 4p = 8 p = 2 & 2p =4 Axis // y-axis opens upward or downward

V = (2 , 4) F = (2 , 4-2) = (2, 2) L = (2 + 4 ,2) = (6 , 2) directrix : y =4+2 or y = 6 Opens downward V = (2 , 4) F = (2 , 4-2) = (2, 2) V = (2 , 4) L = (2 + 4 ,2) = (6 , 2) R = (-2 , 2) L = (6 , 2) R = (2- 4 ,2) = (-2 , 2) F = (2 ,2) To find y-intercept, let x =0 axis : x = 2 To find x-intercept, let y =0

V = (-5, -3) 4p = 12 axis // x-axis p = 3 & 2p = 6 2. V = (-5, -3) 4p = 12 axis // x-axis p = 3 & 2p = 6 opens to the left or right

V = (-5 , -3) F = (-5+3 , -3) = (-2, -3) L = (-2 ,-3+6) = (-2 , 3) Opens to the right V = (-5 , -3) F = (-5+3 , -3) = (-2, -3) L = (-2 , 3) L = (-2 ,-3+6) = (-2 , 3) R = (-2 ,-3-6) = (-2 , -9) To find x-intercept, let y =0 V = (-5 ,-3) F = (-2 ,-3) axis : y = - 3 To find y-intercept, let x =0 R = (-2 , -9) directrix : x =- 5- 3 or x = - 8

Exercises Find the focus, vertex the focal distance, the latus rectum, and the directrix of the parabolas with the following equations. 𝑥 2 =24𝑦 (𝑦−4) 2 =16 𝑥+3 𝑥 2 +20𝑦+4𝑥−60 =0 𝑦 2 −2𝑦−12𝑥=−25

Exercises Find the equation of the parabola which satisfies the given conditions Vertex at (0,0) opens upward with length of latus rectum equal to 8 Focus at (−2,5) and vertex at (2,5) End points of latus rectum at (4,−1) and (4,5) opening the left. Vertex at (0,5) and directrix 𝑥=−4

ELLIPSE is the set of all points P such that the sum of its distances from two fixed points is the same. P -- Foci is constant C -- Center -- vertices -- Major axis L2 L1 -- Minor axis b c C c a = C to endpoints of major axis a a b = C to endpoints of minor axis b R2 R1 c = C to each of the foci L1R1 & L2R2 – latera recta For ellipse, length 2b2/a

Consider an ellipse with center at C(h,k) and major axis parallel to x-axis P(x,y) c c C c a Take the point on the ellipse By def. of ellipse constant sum

STANDARD EQUATIONS OF ELLIPSE Center (h, k) b a b C Major axis // x-axis horizontal C Major axis // y-axis vertical a > b a = b CIRCLE

General Form of Equation of Ellipse A and B have the same signs a b=a A = B CIRCLE

1. Examples C =(-1, 2) Major axis // y-axis : vertical a2 = 25 a = 5 b2 = 4 b = 2 c2 = 25- 4 = 21 c =

A1 b = 2 c = a = 5 C = (-1, 2) A1=(-1, 2+5) = (-1, 7) R1 L1 F1 C = (-1, 2) A1=(-1, 2+5) = (-1, 7) A 2=(-1, 2-5) = (-1, - 3) C B2 B1 B 1=(-1+2, 2) = (1, 2) B 2=(-1-2, 2) = (- 3, 2) F1= L1= F2 R2 L2 R1 = A2 F2 = L2= R2 = Major axis : x = -1 Minor axis : y = 2

2. C =(- 2, 1) Major axis // x-axis : horizontal a2 = 25 a = 5 b2 = 16 b = 4 c2 = 25- 16 = 9 c = 3

b = 4 c = 3 a = 5 C = (-2, 1) B1 A1=(-2+5, 1) = (3, 1) L2 L1 A 2=(-2- 5, 1) = (-7,1) B 1=(-2, 1+4) = (-2, 5) B 2=(-2, 1- 4) = (-2, -3) F2 C F1 A2 A1 F1= (-2+3, 1) = (1,1) L1= R1 = R2 R1 F2 = (-2-3, 1) = (-5, 1) B2 L2= R2 = Major axis : y = 1 Minor axis : x = - 2

3. The only point in the plane that will satisfy the equation is (3 ,0). C(3, 0) The equation represents a point ellipse. Its graph is the point C(3, 0) which is also the center of the ellipse.

4. This is negative. This expression will always result to something positive. Degenerate case of ellipse There exists no point in the plane with real value coordinates x and y that will satisfy this equation thus it has NO GRAPH.

Hyperbola is the set of all points P such that the difference of its distances from two fixed points is the same or constant. --- foci . P is constant C -- Center . . - vertices A1 & A2 -- Transverse axis a = C to endpoints of transverse axis . . L1 -- Conjugate axis L2 . b = C to endpoints of conjugate axis b b2/a b2/a . . . . . . a C a c = C to foci F2 F1 c c L1R1 & L2R2 – latera recta length 2b2/a b b2/a . b2/a . . For Hyperbola, R2 R1 28

ASYMPTOTES OF HYPERBOLA Draw a rectangle with center at C, and of dimensions 2a by 2b . b . . . . Lines through center C and opposite vertices of the rectangle (or the diagonals of the rectangle) are the asymptotes of the hyperbola. . C a a F2 F1 . b

. . A1F2 - A1F1 PF2 – PF1 = 2a - (c - a) (a + c) = 2 a Consider a hyperbola with center at C(h,k) and transverse axis parallel to x-axis Take the point A1 on the hyperbola . . P(x,y) A1F2 - A1F1 (a + c) - (c - a) = 2 a A2 C=(h ,k) A1 Constant difference F2=(h-c ,k) a F1=(h+c, k) C C Take arbitrary point P(x, y) on the hyperbola PF2 – PF1 = 2a By definition of hyperbola,

Following the same algebraic procedure as we did in deriving equation of ellipse, The above equation will result to the following form: For hyperbola, a > , < , = b a2 is together with the positive term, TA // axis of that variable Hyperbola with center at C(h, k), transverse axis // x-axis

STANDARD EQUATIONS OF HYPERBOLA Center (h, k) Transvese axis // x-axis (horizontal) Transverse axis // yaxis (vertical) a > , < , = b

General Form of Equation of Hyperbola A and B have opposite signs

1. . . . . . . . . . . . transverse axis // x-axis a2 = 9 a = 3 b2 = 16 b = 4 c2 = 9 +16 = 25 c = 5 C = (0, 0) A1=(0+3 , 0) = (3, 0) . . A 2=(0- 3, 0) = (-3,0) L2 . L1 B 1= (0, 0+4) = (0, 4) B1 B 2= (0, 0- 4) = (0, - 4) F1= (0+5, 0) = (5,0) . . . . . F2 A2 C A1 F1 L1= R1 = . F2 = (0-5, 0) = (-5, 0) . . B2 R2 R1 L2= R2 = transverse axis : y = 0 conjugate axis : x = 0

2.

. . . . . . . . . . . transverse axis // y-axis F1= F2 = a2 = 9 a = 3 b2 = 4 b = 2 c2 = 9 + 4= 13 c = C = (1, 3) A1=(1, 3+3) = (1, 6) A 2=(1, 3-3 ) = (1,0) B 1= (1+2, 3) = (3, 3) . . . . F1 B 2= (1-2, 3) = (-1, 3) R1 L1 A1 F1= . . . C L1= B2 B1 R1 = . . . A2 . F2 = R2 L2 F2 L2= R2 = transverse axis : x = 1 conjugate axis : y = 3

3. . . . C = (4, 0) degenerate case of hyperbola Equations of intersecting lines representing the asymptotes of the hyperbola

. . . . . . . . . . . equations : asymptotes F1= (5,0) F2 = (-5, 0) C = (0, 0) A1 = (3, 0) . . A 2= (-3,0) L2 . L1 B 1= (0, 4) B1 B 2= (0, - 4) equations : asymptotes F1= (5,0) . . . . . F2 A2 C A1 F1 L1= R1 = F2 = (-5, 0) . . L2= . B2 R2 R1 R2 = transverse axis : y = 0 conjugate axis : x = 0

. . . . . . . . . . . equations : asymptotes F1= F2 = C = (1, 3) A1= (1, 6) A 2= (1,0) B 1= (3, 3) . . . . F1 B 2= (-1, 3) R1 L1 A1 equations : asymptotes F1= . . . C L1= B2 B1 R1 = . . . A2 . F2 = R2 L2 F2 L2= R2 = transverse axis : x = 1 conjugate axis : y = 3

These are called equilateral hyperbolas. If a = b, we have A1 b a A2 C =(h,k) A1 C =(h,k) a b A2 or These are called equilateral hyperbolas.

Equilateral hyperbola with axes rotated are known as rectangular hyperbola. A1 A1 C =(h,k) C =(h,k) A2 A2

X -7 3 6 Y - 5 5 equations : asymptotes x = 2 ; y = - 4 y = - 4 x = 2 C = (2, -4) A1 = ( 2 + 3, -4 + 3 ) = ( 5, -1 ) A 2 = ( 2 - 3 , - 4 - 3) = (- 1, - 7 ) X -7 3 6 Y - 5 5 A1 C equations : asymptotes x = 2 ; y = - 4 y = - 4 A2 x = 2

X -2 3 6 Y - 4 - 9 - 6 x = 2 y = - 5 equations : asymptotes C = (2, - 5) x = 2 A1 = ( 2 - 2, - 5 + 2 ) = ( 0, - 3 ) A 2 = ( 2 + 2 , - 5 - 2) = ( 4, - 7 ) A1 X -2 3 6 Y - 4 - 9 - 6 y = - 5 C A2 equations : asymptotes x = 2 ; y = - 5

Identify & graph the following conic curves 1. 2. 3. 4.