Respuestas
SOLUCIÓN :
1) ( x + 5)²/81 + ( y -3 )²/100 = 1 elipse
centro : C= ( -5 , 3 )= (h,k)
a²= 100 a = 10
b²= 81 b = 9
c² = a²-b² = 100 -81 = 19
c =√19
vértices : A1 = ( h , k+a ) = ( -5 , 3 + 10 )= ( - 5 , 13 )
A2 = ( h , k-a ) = (-5, 3-10) = ( -5 , -7)
B1 = ( h +b , k ) = ( -5+9,3 )= ( 4, 3 )
B2 = ( h -b , k) = ( - 5 -9 , 3 )=( -14 ,3 )
longitud del eje menor =2*a = 2*10= 20
longitud del eje mayor = 2*b = 2*9 = 18
2) 9x²+ 4y²-36x -8y+4 =0
9x² -36x + 4y² -8y = -4
9*( x² -4x + 4 ) + 4*( y²-2y + 1 )= -4 + 36 +4
9*( x-2)²+ 4*( y -1 )² = 36 ÷36
( x -2 )²/ 4 + ( y -1 )²/ 9 = 1
Centro = C= ( h , k) = ( 2 , 1 )
a² = 9 a = 3
b² = 4 = b= 2
a²= b²+ c² c = √( a²-b²)= √( 9-4 )= √5
3 ) 6y²-12x =0
6y²= 12x
y² = 12x/6
y² = 2x ecuación reducida V( 0,0) h=0 k=0
y² = 4py 4p= 2 p = 1/2
foco = ( h +p, k) = ( 1/2 ,0)
directriz = x = -p/2 = x = -1/4
4) ( y +3 )²= 16*( x+ 5 )
Vértice = V= ( h, k ) = ( -5 , -3 )
longitud de lado recto = Lr=I 4*p I = I 4*4 I = 16 .
4p =16 p = 4
5 ) x²/4 - y²/16 = 1
vértices =? focos =? asíntotas =? hipérbola .
a²= 4 a = 2
b²=16 b = 4
c²= a²+b²
c² = 4 + 16 = 20
c = √20
A1 = ( a,0 ) = ( 2 ,0 )
A2 = ( -a,0) = ( -2,0 )
B1 =( 0,b) =( 0, 4 )
B2 = ( 0,-b )= ( 0 , -4 )
F1 =( c ,0)=( √20 ,0 )
F2 = ( -c,0 ) = ( -√20 , 0 )
Asíntotas : y = +- b/a x
y = +- ( 4/2 )x = -+ 2x
y = 2x , y = -2x .