FORMULARIO DE GEOMETRIA ANALITICA

[pic 1]      C(0, 0)

x2 + y2 = r2        e = 0

y2 = 4px        e = 1 V(0, 0) F(p, 0)

x = -p        L.R. = |4p|

[pic 2]   x2 = 4py        e = 1

V(0, 0)        F(0, p)

y = - p        L.R. = |4p|

C(0, 0)

x 2 + y2 =

[pic 3]        [pic 4]

1

a2        b2

V’(-a, 0) V(a, 0)

F’(-c, 0) F(c, 0)

B’(0, -b) B(0, b)

2b2

LR =

a

a2 = b2 + c2

e = c < 1 a

[pic 5]               C(0, 0)

x2        y2

         +                = 1 b2                a2

V’(0, -a) V(0, a)

F’(0, -c) F(0, c)

B’(-b, 0) B(b, 0)

2b2

LR =

a

a2 = b2 + c2

e = c < 1 a

[pic 6]  C(h, k)        e = 0 (x – h)2 + (y – k)2 = r2 x2 + y2 + Dx + Ey + F = 0

D        E

h = −        k = −

2        2

r = 1  D2 + E2 − 4F

2

x2 ± bx = (x ± b )2 − (b )2

2        2

[pic 7]        v(h, k)

(y – k)2 = 4p(x – h) F(h+p, k)

x = h-p

L.R. = |4p|        e = 1 y2+Dx+Ey+F=0

ax ± b= a(𝒙 ± 𝒃)

𝒂

v(h, k) (x – h)2 = 4p(y – k) F(h, k+p)

y = k-p

L.R. = |4p|        e = 1 x2+Dx+Ey+F=0

[pic 8]       C(h, k)

(x − h)2        (y − k)2

+        = 1

a2        b2

V’(h-a, k) V(h+a, k)

F’(h-c, k) F(h+c, k)

B’(h, k-b)  B(h, k+b)

2b2

LR =

a

e = c < 1 a

a2 = b2 + c2

Ax2+Cy2+Dx+Ey+F=0

C(h, k)

(x − h)2 + (y − k)2 =

b2        a2        1

V’(h, k-a) V(h, k+a)

F’(h, k-c) F(h, k+c)

B’(h-b, k) B(h+b, k)

2b2

LR =

a

e = c < 1 a

a2 = b2 + c2

Ax2+Cy2+Dx+Ey+F=0

C(0, 0)

x2        y2

          −                = 1 a2                b2

V’(-a, 0) V(a, 0)

F’(-c, 0)        F(c, 0)

B’(0 –b) B(0, b)

2b2

LR =

a

c2 = a2 + b2

e = c > 1 a

y = ± b x

a

Ax2+Cy2+F=0

C(0, 0)

y2        x2

          −                = 1 a2                b2

V’(0, -a) V(0, a)

F’(0, -c)        F(0, c)

B’(-b, 0) B(b, 0)

2b2

LR =

a

c2 = a2 + b2

e = c > 1 a

y = ± a x

b

Ax2+Cy2+F=0

C(h, k)

(x − h)2 − (y − k)2 =

1

a2        b2

V’(h-a, k) V(h+a, k)

F’(h-c, k) F(h+c,k)

B’(h, k-b) B(h, k+b)

2b2

LR =

a

c2 = a2 + b2

e = c > 1 a

y = ± b (x-h)+k

a

Ax2+Cy2+Dx+Ey+F=0

d= P1P2= (x2 − x1)2 + (y2 − y1)2

x = x1 + rx2        y = y1 + ry2        r = x − x1 = y − y1

1 + r        1 + r        x2 − x        y2 − y

A = 1 [x1y2 + x2y3 + x3y1 − x1y3 − x3y2 − x2y1]

2

A = 1 [x1(y2 − yn) + x2(y3 − y1) + x3(y4 − y2). . .]

2

xcos(ω)+sen(ω)−p = 0

A x +  B y +  C  = 0        r =   A2 + B2

± r        ± r        ± r

cos(ω) = A        sen(ω) = B        − p = C

± r        ± r        ± r

Ax + By + C = 0, P1(x1, y1)        l1 || l2

d = Ax1 + By1 + C        d =  C1 − C2

±   A2 + B2        A2 + B2

d =  b2 − b1

A2 + B2

± d1 = ±d2

± A1x + B1y + C1 = ± A2 x + B2 y + C2

±    A2 + B2        ±    A2 + B2

1        1        2        2

C(h, k)

(y − k)2 − (x − h)2 =

a2        b2        1

V’(h, k-a) V(h, k+a)

F’(h, k-c) F(h, k+c)

B’(h-b, k) B(h+b, k)

2b2

LR =

a

c2 = a2 + b2

e = c > 1 a

y = ± a (x-h)+k

b

Ax2+Cy2+Dx+Ey+F=0

x                = x1 + x2 m        2

y + y ym  =    1        2

2

x = x1 + r(x2 − x1) y = y1 + r(y2 − y1)

r =  x − x1   =   y − y1  

x2 − x1        y2 − y1

m = y2 − y1

x2 − x1

tanθ = m

tan θ = m2 − m1

1 + m2m1

l1 || l2 ⇒ m1 = m2

l1 ⊥ l2 ⇒ m1m2 = −1

y − y1 = m(x − x1) y = mx + b

y − y = y2 − y1 (x − x )

1        x2 − x1        1

x1        y1

x2        y2

A = 1 ⁝        ⁝

2 x        y

n        n

x1        y1

Ax + By + C = 0

m = − A   b = − C

B        B

x + y = 1 a        b

a = − C        b = − C

A        B

l2 || l1

y − y1 = m1(x − x1)

l2 ⊥ l1

y − y1 = − 1 (x − x1)

m1